Showing 1–50 of 51 results for author: Chatterjee, T

Quadratic Residue Codes over $\mathbb{Z}_{121}$

Authors: Tapas Chatterjee, Priya Jain

Abstract: In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \) and \( p \equiv \pm 19 \pmod{44} \) over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and ex… ▽ More In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \) and \( p \equiv \pm 19 \pmod{44} \) over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and extended quadratic residue codes over $\mathbb{Z}_{121}$ are discussed, followed by their Gray images. Also, we show that the extended quadratic residue code over $\mathbb{Z}_{121}$ possesses a large permutation automorphism group generated by shifts, multipliers, and inversion, making permutation decoding feasible. As examples, we construct new codes with parameters $[55,5,33]$ and $[77,7,44].$ △ Less

Submitted 25 March, 2026; originally announced March 2026.

On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions

Authors: Tapas Chatterjee, Sagar Mandal

Abstract: The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $τ(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-p… ▽ More The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $τ(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-periodic. Further, we deduce that the sequence arising from Nathanson's totient function $Φ(n)$, the classical Euler's totient function $\varphi(n)$, sum of divisor function $σ(n)$, their Dirichlet convolution $σ*\varphi(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $\varphi^*(n)$ modulo $m$, are non-periodic for certain modulo $m$. In addition, we extend a result of Ayad and Kihel \cite{r1} on the non-periodicity of certain arithmetic function $g(n)$. On the other hand, we construct several transcendental numbers arising from the continued fractions attached with $τ(n)$, $E_k(z)$, $Φ(n)$, $g(n)$, $\varphi(n)$, $σ(n)$, $σ*\varphi(n)$, $J_k(n)$, and $\varphi^*(n)$. △ Less

Submitted 9 February, 2026; originally announced February 2026.

Comments: 13 pages

MSC Class: Primary: 11A25; 11F11; 40A15; 11J81; Secondary: 11A55; 11B50

A Note on primitive pairs for graded Lie algebras

Authors: Tamanna Chatterjee

Abstract: We develop a theory of primitive pairs for $\mathbb{Z}$-graded Lie algebras when the sheaves have coefficients in a field $\Bbbk$ of positive characteristic, providing a graded analogue of the role played by cuspidal pairs in the generalized Springer correspondence. We consider the centralizer $G_0$ of a fixed cocharacter $χ$ in a connected, reductive, algebraic group $G$ and its action on the eig… ▽ More We develop a theory of primitive pairs for $\mathbb{Z}$-graded Lie algebras when the sheaves have coefficients in a field $\Bbbk$ of positive characteristic, providing a graded analogue of the role played by cuspidal pairs in the generalized Springer correspondence. We consider the centralizer $G_0$ of a fixed cocharacter $χ$ in a connected, reductive, algebraic group $G$ and its action on the eigenspaces $\mathfrak{g}_n$ of $χ$. Building on the framework of parity sheaves and the Fourier transform established in \cite{Ch,Ch1}, we show that every indecomposable parity sheaf on $\mathfrak{g}_n$ can be expressed as a direct summand of a complex induced from primitive data on a Levi subgroup. This result extends the fact that, in the graded setting, any indecomposable parity sheaf is direct summand of an induced cuspidal datum \cite{Ch}. This confirms the organizing role of primitive pairs in the block decomposition of the category of $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$. We further establish that primitive pairs on the nilpotent cone induce primitive pairs in the graded setting, and we prove that primitivity is preserved under the Fourier--Sato transform. These results reveal a deep compatibility between the geometry of graded Lie algebras and their representation-theoretic structures, forming the foundation for a graded version of the generalized Springer correspondence in positive characteristic. △ Less

Submitted 28 October, 2025; originally announced October 2025.

Comments: 19 pages

Modified security analysis of device-independent quantum key distribution with random key basis

Authors: Sawan Bhattacharyya, Turbasu Chatterjee, Pankaj Agrawal, Prasenjit Deb

Abstract: Security analysis is a critical part in any cryptographic protocol, may it be classical or quantum. Without security analysis, one cannot ensure the secrecy of the distributed keys. To perform a conclusive security analysis, it is very often necessary to frame the problem as an optimization problem. However, solving such optimization problems is quite challenging. In this article, we focus on the… ▽ More Security analysis is a critical part in any cryptographic protocol, may it be classical or quantum. Without security analysis, one cannot ensure the secrecy of the distributed keys. To perform a conclusive security analysis, it is very often necessary to frame the problem as an optimization problem. However, solving such optimization problems is quite challenging. In this article, we focus on the security analysis of device-independent quantum key distribution (DIQKD) with random key basis protocol. We show that the optimization cost of the existing security analysis can be reduced without compromising the key rate. In particular, we reframe the entire security analysis of this protocol as a strongly convex optimization problem and demonstrate that unlike the original security proof, optimization of Bob's measurement angles for finding a lower bound on Eve's uncertainty about Alice's key generation basis can be done with lesser cost. We derive an explicit form of the pessimistic error that arises while optimizing the measurement angles of both the parties. We also clarify a few parts of the original security proof, making the analysis more rigorous and complete. △ Less

Submitted 18 August, 2025; originally announced August 2025.

Comments: 11+38 pages, comments and suggestions are welcome

Distribution of Farey fractions with $k$-free denominators

Authors: Bittu Chahal, Tapas Chatterjee, Sneha Chaubey

Abstract: We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues o… ▽ More We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($k\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($k=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains. △ Less

Submitted 3 July, 2025; v1 submitted 30 June, 2025; originally announced July 2025.

MSC Class: 11B57 \sep 11J71 \sep 11K38 \sep 11L07 \sep 11L15 \sep 11M26

Equivariant sheaves for classical groups acting on Grassmannians

Authors: Pramod N. Achar, Tamanna Chatterjee

Abstract: Let $V$ be a finite-dimensional complex vector space. Assume that $V$ is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this paper, we introduce a stratification of the Grassmannian $\mathrm{Gr}_k(V)$ related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratif… ▽ More Let $V$ be a finite-dimensional complex vector space. Assume that $V$ is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this paper, we introduce a stratification of the Grassmannian $\mathrm{Gr}_k(V)$ related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification. The main results involve sheaves with coefficients in a field of characteristic other than $2$. We prove that there are "enough" parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property. This situation arises in the following context: let $x$ be a nilpotent element in the Lie algebra of either $G = \mathrm{Sp}_N(\mathbb{C})$ or $G = \mathrm{SO}_N(\mathbb{C})$, and let $V = \ker x \subset \mathbb{C}^N$. Our stratification of $\mathrm{Gr}_k(V)$ is preserved by the centralizer $G^x$, and we expect our results to have applications in Springer theory for classical groups. △ Less

Submitted 9 April, 2025; v1 submitted 5 November, 2024; originally announced November 2024.

Comments: 33 pages. v2: added results on tangent spaces and transverse slices

Fast classical simulation of qubit-qudit hybrid systems

Authors: Haemanth Velmurugan, Arnav Das, Turbasu Chatterjee, Amit Saha, Anupam Chattopadhyay, Amlan Chakrabarti

Abstract: Simulating quantum circuits is a computationally intensive task that relies heavily on tensor products and matrix multiplications, which can be inefficient. Recent advancements, eliminate the need for tensor products and matrix multiplications, offering significant improvements in efficiency and parallelization. Extending these optimizations, we adopt a block-simulation methodology applicable to q… ▽ More Simulating quantum circuits is a computationally intensive task that relies heavily on tensor products and matrix multiplications, which can be inefficient. Recent advancements, eliminate the need for tensor products and matrix multiplications, offering significant improvements in efficiency and parallelization. Extending these optimizations, we adopt a block-simulation methodology applicable to qubit-qudit hybrid systems. This method interprets the statevector as a collection of blocks and applies gates without computing the entire circuit unitary. Our method, a spiritual successor of the simulator QuDiet \cite{Chatterjee_2023}, utilizes this block-simulation method, thereby gaining major improvements over the simulation methods used by its predecessor. We exhibit that the proposed method is approximately 10$\times$ to 1000$\times$ faster than the state-of-the-art simulator for simulating multi-level quantum systems with various benchmark circuits. △ Less

Submitted 23 October, 2024; originally announced October 2024.

Comments: 12 pages, 10 figures

Stochastic Nonlinear Model Updating in Structural Dynamics Using a Novel Likelihood Function within the Bayesian-MCMC Framework

Authors: Pushpa Pandey, Hamed Haddad Khodaparast, Michael Ian Friswell, Tanmoy Chatterjee, Hadi Madinei, Tom Deighan

Abstract: The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its c… ▽ More The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated. △ Less

Submitted 14 October, 2024; originally announced October 2024.

On Characterizing Potential Friends of 20

Authors: Tapas Chatterjee, Sagar Mandal, Sourav Mandal

Abstract: Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then… ▽ More Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $Ω(n)$ and $ω(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$. △ Less

Submitted 15 September, 2025; v1 submitted 25 August, 2024; originally announced September 2024.

Comments: 25 pages

MSC Class: 11A25

Journal ref: Annals of West University of Timisoara - Mathematics and Computer Science, 61(1), 2025. 205-229

On characterization of prime divisors of the index of a quadrinomial

Authors: Tapas Chatterjee, Karishan Kumar

Abstract: Let $θ$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $θ$ over the rationals. Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x).$ As a fascinating corollary, we deduce necessary and sufficient… ▽ More Let $θ$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $θ$ over the rationals. Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x).$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(θ),$ where $θ$ is associated with certain quadrinomials. △ Less

Submitted 7 January, 2025; v1 submitted 26 August, 2024; originally announced August 2024.

Comments: 18 pages

MSC Class: Primary: 11R04; 11R29; 11Y40; Secondary: 11R09; 11R21

On characterization of Monogenic number fields associated with certain quadrinomials and its applications

Authors: Tapas Chatterjee, Karishan Kumar

Abstract: Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $θ$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $θ.$ As an interesting result, we esta… ▽ More Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $θ$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $θ.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(θ)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$ △ Less

Submitted 7 January, 2025; v1 submitted 26 August, 2024; originally announced August 2024.

Comments: 23 pages

MSC Class: Primary: 11R04; 11R29; 11Y40; Secondary: 11R09; 11R21

Special values of derivatives of certain $L$-functions

Authors: Tapas Chatterjee, Sonika Dhillon

Abstract: In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcend… ▽ More In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where $L'(0,f)$ is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of $L$-functions. △ Less

Submitted 25 August, 2024; originally announced August 2024.

Comments: 12 pages

MSC Class: 11J81; 11J86; 11M06; 11J91

A note on MDS Property of Circulant Matrices

Authors: Tapas Chatterjee, Ayantika Laha

Abstract: In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS mat… ▽ More In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in $2022,$ Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field $\mathbb{F}_{2^m}.$ Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic $2$, the trace of associated diagonal matrices may possess non-zero values. △ Less

Submitted 22 June, 2024; originally announced June 2024.

Comments: 15 pages. arXiv admin note: text overlap with arXiv:2406.15872

MSC Class: 12E20; 15B10; 94A60 (Primary); 15B05 ( Secondary)

On MDS Property of g-Circulant Matrices

Authors: Tapas Chatterjee, Ayantika Laha

Abstract: Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In… ▽ More Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In $2016$, Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In $1961,$ Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss $g$-circulant matrices with involutory and MDS properties. We prove that $g$-circulant involutory matrices of order $k \times k$ cannot be MDS unless $g \equiv -1 \pmod k.$ Next, we delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the $k$-th power of the associated diagonal matrices of a $g$-circulant semi-orthogonal (semi-involutory) matrix of order $k \times k$ results in a scalar matrix. These findings can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {\it{et al.}} in $2022.$ △ Less

Submitted 22 June, 2024; originally announced June 2024.

Comments: 21 pages

MSC Class: 12E20; 15B10; 94A60 (Primary); 15B05 ( Secondary)

A note on cyclic MDS and non-MDS matrices

Authors: Tapas Chatterjee, Ayantika Laha

Abstract: In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the e… ▽ More In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to $g$-circulant matrices. Additionally, we determine the determinant of $g$-circulant matrices of order $2^d \times 2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic $2$. Furthermore, we prove that this result holds for any cyclic matrix. △ Less

Submitted 7 January, 2025; v1 submitted 20 June, 2024; originally announced June 2024.

Comments: 18 pages

MSC Class: 94A60; 12E20; 15B10 (Primary); 15B05 (Secondary)

A Characterization of Semi-Involutory MDS Matrices

Authors: Tapas Chatterjee, Ayantika Laha

Abstract: In symmetric cryptography, maximum distance separable (MDS) matrices with computationally simple inverses have wide applications. Many block ciphers like AES, SQUARE, SHARK, and hash functions like PHOTON use an MDS matrix in the diffusion layer. In this article, we first characterize all $3 \times 3$ irreducible semi-involutory matrices over the finite field of characteristic $2$. Using this matr… ▽ More In symmetric cryptography, maximum distance separable (MDS) matrices with computationally simple inverses have wide applications. Many block ciphers like AES, SQUARE, SHARK, and hash functions like PHOTON use an MDS matrix in the diffusion layer. In this article, we first characterize all $3 \times 3$ irreducible semi-involutory matrices over the finite field of characteristic $2$. Using this matrix characterization, we provide a necessary and sufficient condition to construct MDS semi-involutory matrices using only their diagonal entries and the entries of an associated diagonal matrix. Finally, we count the number of $3 \times 3$ semi-involutory MDS matrices over any finite field of characteristic $2$. △ Less

Submitted 1 January, 2025; v1 submitted 18 June, 2024; originally announced June 2024.

Comments: 19 pages

MSC Class: 05B20; 12E20; 15B99 (Primary); 94A60; 94B05 (Secondary)

Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields

Authors: Tapas Chatterjee, Sonam Garg

Abstract: In this article, we aim to extend the research conducted by Chatterjee and Garg in 2024, particularly focusing on the $q$-analogue of the generalized Stieltjes constants. These constants constitute the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. Chatterjee and Garg previously established arithmetic results related to $γ_0(q,x)$, for… ▽ More In this article, we aim to extend the research conducted by Chatterjee and Garg in 2024, particularly focusing on the $q$-analogue of the generalized Stieltjes constants. These constants constitute the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. Chatterjee and Garg previously established arithmetic results related to $γ_0(q,x)$, for $q>1$ and $0 < x <1$ over the field of rational numbers. Here, we broaden their findings to encompass number fields $\mathbb{F}$ in two scenarios: firstly, when $\mathbb{F}$ is linearly disjoint from the cyclotomic field $\mathbb{Q}(ζ_b)$, and secondly, when $\mathbb{F}$ has non-trivial intersection with $\mathbb{Q}(ζ_b)$, with $b \geq 3$ being any positive integer. △ Less

Submitted 11 April, 2024; originally announced April 2024.

Comments: arXiv admin note: substantial text overlap with arXiv:2404.08025

On arithmetic nature of $q$-analogue of the generalized Stieltjes constants

Authors: Tapas Chatterjee, Sonam Garg

Abstract: In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laur… ▽ More In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laurent series of the $q$-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to $γ_0(q,x)$ for $q>1$ and $0 < x <1$, where $γ_0(q,x)$ is the constant term which appears in the Laurent series expansion of $q$-Hurwitz zeta function around $s=1$. Furthermore, we put forth a conjecture about the linear independence of special values of $γ_0(q,x)$ along with $1$ at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals. △ Less

Submitted 11 April, 2024; originally announced April 2024.

Transcendental nature of $p$-adic digamma values

Authors: Tapas Chatterjee, Sonam Garg

Abstract: For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $ψ_p(r/p)+ γ_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $ψ_p(r/p^n)+ γ_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental n… ▽ More For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $ψ_p(r/p)+ γ_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $ψ_p(r/p^n)+ γ_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $ψ_p(r/pq)+ γ_p$, where $p, q$ are distinct primes. △ Less

Submitted 11 April, 2024; originally announced April 2024.

Algebraic identities among $q$- analogue of Euler double zeta values

Authors: Tapas Chatterjee, Sonam Garg

Abstract: In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant. Using a $q$-analogue of the Nielsen Reflexion Formula for $q>1$, we investigate identities involving different versions of $q$-analogues of the Riemann zeta functi… ▽ More In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant. Using a $q$-analogue of the Nielsen Reflexion Formula for $q>1$, we investigate identities involving different versions of $q$-analogues of the Riemann zeta function and the double-zeta function. Additionally, we analyze the behavior of $ζ_q(s_1, s_2)$ as $s_1$ and $s_2$ approach to $0$ and compare these limits to those of the classical double-zeta function. Finally, we discuss the $q$-analogue of the Mordell-Tornheim $r$-ple zeta function and its relation with the $q$-double zeta function. △ Less

Submitted 11 April, 2024; originally announced April 2024.

A note on necessary conditions for a friend of 10

Authors: Tapas Chatterjee, Sagar Mandal, Sourav Mandal

Abstract: Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ wh… ▽ More Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ where $f$ is the smallest odd positive integer greater than $1$ and less than or equal to $\min\{ 2a+1,p-1\}$, provided $5^{2a}\mid \mid N$. Further, there exist prime factors $p$ and $q$ (not necessarily distinct) of $N$ such that $p\equiv1 \pmod {10}$ and $q\equiv 1\pmod 6$. Besides, we prove that if a Fermat prime $F_k$ divides $N$ then $N$ must have a prime factor congruent to $1$ modulo $2F_k$. Also, if we consider the form of $N$ as $N=5^{2a}m^2$ then $m$ is non square-free. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-4$ and if $Ω(m)\leq K$ then $N< 5\cdot 6^{(2^{K-2a+1}-1)^2}$ where $Ω(n)$ and $ω(n)$ denote the total number of prime factors and the number of distinct prime factors of the integer $n$ respectively. △ Less

Submitted 16 January, 2025; v1 submitted 31 March, 2024; originally announced April 2024.

MSC Class: 11Axx; 11A25

Action of the relative Weyl group on partial Springer sheaf

Authors: Tamanna Chatterjee, Laura Rider

Abstract: In the context of the Springer correspondence, the Weyl group action on the Springer sheaf can be defined in two ways: via restriction or the Fourier transform. It is well-known that these two actions differ by the sign character. This was proven by Hotta for sheaves with characteristic 0 coefficients in 1981, and more recently extended to arbitrary coefficients by Achar, Henderson, Juteau, and Ri… ▽ More In the context of the Springer correspondence, the Weyl group action on the Springer sheaf can be defined in two ways: via restriction or the Fourier transform. It is well-known that these two actions differ by the sign character. This was proven by Hotta for sheaves with characteristic 0 coefficients in 1981, and more recently extended to arbitrary coefficients by Achar, Henderson, Juteau, and Riche in 2014. In this short article, we study an extension of this problem to the partial Springer sheaf with arbitrary field coefficients. This involves an action of the so-called relative Weyl group $W(L)$. △ Less

Submitted 27 March, 2024; originally announced March 2024.

Comments: 18 pages

MSC Class: 18G80

Charge-transfer-driven enhanced room-temperature ferromagnetism in BiFeO$_3$/Ag nanocomposite

Authors: Tania Chatterjee, Shubhankar Mishra, Arnab Mukherjee, Prabir Pal, Biswarup Satpati, Dipten Bhattacharya

Abstract: We report observation of more than an order of magnitude jump in saturation magnetization in BiFeO$_3$/Ag nanocomposite at room temperature compared to what is observed in bare BiFeO$_3$ nanoparticles. Using transmission electron microscopy together with energy dispersive x-ray spectra (which maps the element concentration across the BiFeO$_3$/Ag interface) and x-ray photoelectron spectroscopy, we… ▽ More We report observation of more than an order of magnitude jump in saturation magnetization in BiFeO$_3$/Ag nanocomposite at room temperature compared to what is observed in bare BiFeO$_3$ nanoparticles. Using transmission electron microscopy together with energy dispersive x-ray spectra (which maps the element concentration across the BiFeO$_3$/Ag interface) and x-ray photoelectron spectroscopy, we show that both the observed specific self-assembly pattern of BiFeO$_3$ and Ag nanoparticles and the charge transfer between Ag and O are responsible for such an enormous rise in room-temperature magnetization. The BiFeO$_3$/Ag nanocomposites, therefore, could prove to be extremely useful for a variety of applications including biomedical. △ Less

Submitted 6 September, 2023; originally announced September 2023.

Comments: 11 pages, 7 figures, accepted for publication in Nanotechnology

Journal ref: Nanotechnology 34, 495705 (2023)

Beyond Counting Datasets: A Survey of Multilingual Dataset Construction and Necessary Resources

Authors: Xinyan Velocity Yu, Akari Asai, Trina Chatterjee, Junjie Hu, Eunsol Choi

Abstract: While the NLP community is generally aware of resource disparities among languages, we lack research that quantifies the extent and types of such disparity. Prior surveys estimating the availability of resources based on the number of datasets can be misleading as dataset quality varies: many datasets are automatically induced or translated from English data. To provide a more comprehensive pictur… ▽ More While the NLP community is generally aware of resource disparities among languages, we lack research that quantifies the extent and types of such disparity. Prior surveys estimating the availability of resources based on the number of datasets can be misleading as dataset quality varies: many datasets are automatically induced or translated from English data. To provide a more comprehensive picture of language resources, we examine the characteristics of 156 publicly available NLP datasets. We manually annotate how they are created, including input text and label sources and tools used to build them, and what they study, tasks they address and motivations for their creation. After quantifying the qualitative NLP resource gap across languages, we discuss how to improve data collection in low-resource languages. We survey language-proficient NLP researchers and crowd workers per language, finding that their estimated availability correlates with dataset availability. Through crowdsourcing experiments, we identify strategies for collecting high-quality multilingual data on the Mechanical Turk platform. We conclude by making macro and micro-level suggestions to the NLP community and individual researchers for future multilingual data development. △ Less

Submitted 28 November, 2022; originally announced November 2022.

Comments: Accepted to Findings of EMNLP 2022. You can view our annotations, contribute to our survey, and view the analysis visualizations on our website at https://multilingual-dataset-survey.github.io

Fourier transform on graded Lie algebras

Authors: Tamanna Chatterjee

Abstract: In this paper we study the Fourier transform on graded Lie algebras. Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and also assuming two conjectures for the gr… ▽ More In this paper we study the Fourier transform on graded Lie algebras. Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and also assuming two conjectures for the group $G$, we prove that the Fourier transform sends parity complexes to parity complexes. Primitive pairs have played an important role in Lusztig's paper \cite{Lu} to prove a block decomposition in the graded setting. A long term goal of this project is to prove a similar block decomposition in positive characteristic. In this paper we have tried to understand the primitive pair and its relation with the Fourier transform. △ Less

Submitted 30 September, 2025; v1 submitted 25 November, 2022; originally announced November 2022.

Comments: Removed two sections, minor changes, 18 pages

QuDiet: A Classical Simulation Platform for Qubit-Qudit Hybrid Quantum Systems

Authors: Turbasu Chatterjee, Arnav Das, Subhayu Kumar Bala, Amit Saha, Anupam Chattopadhyay, Amlan Chakrabarti

Abstract: In the recent years, numerous research advancements have extended the limit of classical simulation of quantum algorithms. Although, most of the state-of-the-art classical simulators are only limited to binary quantum systems, which restrict the classical simulation of higher-dimensional quantum computing systems. Through recent developments in higher-dimensional quantum computing systems, it is r… ▽ More In the recent years, numerous research advancements have extended the limit of classical simulation of quantum algorithms. Although, most of the state-of-the-art classical simulators are only limited to binary quantum systems, which restrict the classical simulation of higher-dimensional quantum computing systems. Through recent developments in higher-dimensional quantum computing systems, it is realized that implementing qudits improves the overall performance of a quantum algorithm by increasing memory space and reducing the asymptotic complexity of a quantum circuit. Hence, in this article, we introduce \textbf{QuDiet}, a state-of-the-art user-friendly python-based higher-dimensional quantum computing simulator. \textbf{QuDiet} offers multi-valued logic operations by utilizing generalized quantum gates with an abstraction so that any naive user can simulate qudit systems with ease as compared to the existing ones. We simulate various benchmark quantum circuits in \textbf{QuDiet} and show the considerable speedup in simulation time as compared to the other simulators without loss in precision. Finally, \textbf{QuDiet} provides a full qubit-qudit hybrid quantum simulator package with quantum circuit templates of well-known quantum algorithms for fast prototyping and simulation. The complete code and packages of \textbf{QuDiet} is available at https://github.com/LegacYFTw/QuDiet so that other platforms can incorporate it as a classical simulation option for qubit-qudit hybrid systems to their platforms. △ Less

Submitted 15 November, 2022; originally announced November 2022.

Comments: 18 pages, 6 figures

Journal ref: IET Quantum Communication (2023)

longhorns at DADC 2022: How many linguists does it take to fool a Question Answering model? A systematic approach to adversarial attacks

Authors: Venelin Kovatchev, Trina Chatterjee, Venkata S Govindarajan, Jifan Chen, Eunsol Choi, Gabriella Chronis, Anubrata Das, Katrin Erk, Matthew Lease, Junyi Jessy Li, Yating Wu, Kyle Mahowald

Abstract: Developing methods to adversarially challenge NLP systems is a promising avenue for improving both model performance and interpretability. Here, we describe the approach of the team "longhorns" on Task 1 of the The First Workshop on Dynamic Adversarial Data Collection (DADC), which asked teams to manually fool a model on an Extractive Question Answering task. Our team finished first, with a model… ▽ More Developing methods to adversarially challenge NLP systems is a promising avenue for improving both model performance and interpretability. Here, we describe the approach of the team "longhorns" on Task 1 of the The First Workshop on Dynamic Adversarial Data Collection (DADC), which asked teams to manually fool a model on an Extractive Question Answering task. Our team finished first, with a model error rate of 62%. We advocate for a systematic, linguistically informed approach to formulating adversarial questions, and we describe the results of our pilot experiments, as well as our official submission. △ Less

Submitted 29 June, 2022; originally announced June 2022.

Comments: Accepted at DADC2022

Intermediate Qutrit-based Improved Quantum Arithmetic Operations with Application on Financial Derivative Pricing

Authors: Amit Saha, Turbasu Chatterjee, Anupam Chattopadhyay, Amlan Chakrabarti

Abstract: In some quantum algorithms, arithmetic operations are of utmost importance for resource estimation. In binary quantum systems, some efficient implementation of arithmetic operations like, addition/subtraction, multiplication/division, square root, exponential and arcsine etc. have been realized, where resources are reported as a number of Toffoli gates or T gates with ancilla. Recently it has been… ▽ More In some quantum algorithms, arithmetic operations are of utmost importance for resource estimation. In binary quantum systems, some efficient implementation of arithmetic operations like, addition/subtraction, multiplication/division, square root, exponential and arcsine etc. have been realized, where resources are reported as a number of Toffoli gates or T gates with ancilla. Recently it has been demonstrated that intermediate qutrits can be used in place of ancilla, allowing us to operate efficiently in the ancilla-free frontier zone. In this article, we have incorporated intermediate qutrit approach to realize efficient implementation of all the quantum arithmetic operations mentioned above with respect to gate count and circuit-depth without T gate and ancilla. Our resource estimates with intermediate qutrits could guide future research aimed at lowering costs considering arithmetic operations for computational problems. As an application of computational problems, related to finance, are poised to reap the benefit of quantum computers, in which quantum arithmetic circuits are going to play an important role. In particular, quantum arithmetic circuits of arcsine and square root are necessary for path loading using the re-parameterization method, as well as the payoff calculation for derivative pricing. Hence, the improvements are studied in the context of the core arithmetic circuits as well as the complete application of derivative pricing. Since our intermediate qutrit approach requires to access higher energy levels, making the design prone to errors, nevertheless, we show that the percentage decrease in the probability of error is significant owing to the fact that we achieve circuit robustness compared to qubit-only works. △ Less

Submitted 31 May, 2022; originally announced May 2022.

Comments: 11 pages, 6 figures

Signal Quality Assessment of Photoplethysmogram Signals using Quantum Pattern Recognition and lightweight CNN Architecture

Authors: Tamaghno Chatterjee, Aayushman Ghosh, Sayan Sarkar

Abstract: Photoplethysmography (PPG) signal comprises physiological information related to cardiorespiratory health. However, while recording, these PPG signals are easily corrupted by motion artifacts and body movements, leading to noise enriched, poor quality signals. Therefore ensuring high-quality signals is necessary to extract cardiorespiratory information accurately. Although there exists several rul… ▽ More Photoplethysmography (PPG) signal comprises physiological information related to cardiorespiratory health. However, while recording, these PPG signals are easily corrupted by motion artifacts and body movements, leading to noise enriched, poor quality signals. Therefore ensuring high-quality signals is necessary to extract cardiorespiratory information accurately. Although there exists several rule-based and Machine-Learning (ML) - based approaches for PPG signal quality estimation, those algorithms' efficacy is questionable. Thus, this work proposes a lightweight CNN architecture for signal quality assessment employing a novel Quantum pattern recognition (QPR) technique. The proposed algorithm is validated on manually annotated data obtained from the University of Queensland database. A total of 28366, 5s signal segments are preprocessed and transformed into image files of 20 x 500 pixels. The image files are treated as an input to the 2D CNN architecture. The developed model classifies the PPG signal as `good' or `bad' with an accuracy of 98.3% with 99.3% sensitivity, 94.5% specificity and 98.9% F1-score. Finally, the performance of the proposed framework is validated against the noisy `Welltory app' collected PPG database. Even in a noisy environment, the proposed architecture proved its competence. Experimental analysis concludes that a slim architecture along with a novel Spatio-temporal pattern recognition technique improve the system's performance. Hence, the proposed approach can be useful to classify good and bad PPG signals for a resource-constrained wearable implementation. △ Less

Submitted 1 February, 2022; originally announced February 2022.

Comments: 7 pages, 6 figures, submitted to IEEE EMBC 2022

Towards 6G Communications: Architecture, Challenges, and Future Directions

Authors: Purbita Mitra, Rouprita Bhattacharjee, Twinkle Chatterjee, Soumalya De, Raja Karmakar, Arindam Ghosh, Tinku Adhikari

Abstract: The cellular network standard is gradually stepping towards the 6th Generation (6G). In 6G, the pioneering and exclusive features, such as creating connectivity even in space and under water, are attracting Governments, organizations and researchers to spend time, money, effort extensively in this area. In the direction of intelligent network management and distributed secured systems, Artificial… ▽ More The cellular network standard is gradually stepping towards the 6th Generation (6G). In 6G, the pioneering and exclusive features, such as creating connectivity even in space and under water, are attracting Governments, organizations and researchers to spend time, money, effort extensively in this area. In the direction of intelligent network management and distributed secured systems, Artificial Intelligence (AI) and blockchain are going to form the backbone of 6G, respectively. However, there is a need for the study of the 6g architecture and technology, such that researchers can identify the scopes of improvement in 6G. Therefore, in this survey, we discuss the primary requirements of 6G along with its overall architecture and technological aspects. We also highlight crucial challenges and future research directions in 6G networks, which can lead to the successful practical implementation of 6G, as per the objective of its introduction in next generation cellular networks. △ Less

Submitted 16 January, 2022; originally announced January 2022.

Journal ref: 2021 12th International Conference on Computing Communication and Networking Technologies (ICCCNT), 2021

Energy Landscape Structure of Small Graph Isomorphism Under Variational Optimization

Authors: Turbasu Chatterjee, Shah Ishmam Mohtashim, Akash Kundu

Abstract: We investigate a quadratic unconstrained binary optimization (QUBO) formulation of the graph isomorphism problem using the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). For small graph instances, we observe that isomorphic pairs exhibit consistent clustering in variational energies, indicating that the Hamiltonian successfully encodes structural f… ▽ More We investigate a quadratic unconstrained binary optimization (QUBO) formulation of the graph isomorphism problem using the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). For small graph instances, we observe that isomorphic pairs exhibit consistent clustering in variational energies, indicating that the Hamiltonian successfully encodes structural features. However, we demonstrate that low variational energy alone is an unreliable certifier of isomorphism due to the high probability of converging to infeasible states that violate bijection constraints. To address this, we analyze optimization trajectories rather than final energies; consistently outperform naive energy thresholding, though absolute performance remains limited. Our results characterize the current limits of variational algorithms for graph isomorphism, positioning energy landscape analysis as a diagnostic tool rather than a scalable decision procedure in the NISQ regime. △ Less

Submitted 14 January, 2026; v1 submitted 18 November, 2021; originally announced November 2021.

Comments: 7 pages, 3 figures

Journal ref: AIP Advances 16, 015322 (2026)

Nonmonotonic magnetic field dependence of remanent ferroelectric polarization in reduced-graphene-oxide-BiFeO$_3$ nanocomposite

Authors: Tania Chatterjee, Arnab Mukherjee, Prabir Pal, S. D. Kaushik, V. Siruguri, Swarupananda Bhattacharjee, Chandan Kumar Ghosh, Dipten Bhattacharya

Abstract: In a nanocomposite of reduced graphene oxide (RGO) and BiFeO$_3$ (BFO), the remanent ferroelectric polarization is found to follow nonmonotonic magnetic field dependence at room temperature as the applied magnetic field is swept across 0-20 kOe on a pristine sample. The remanent ferroelectric polarization is determined both from direct electrical measurements on an assembly of nanoparticles and po… ▽ More In a nanocomposite of reduced graphene oxide (RGO) and BiFeO$_3$ (BFO), the remanent ferroelectric polarization is found to follow nonmonotonic magnetic field dependence at room temperature as the applied magnetic field is swept across 0-20 kOe on a pristine sample. The remanent ferroelectric polarization is determined both from direct electrical measurements on an assembly of nanoparticles and powder neutron diffraction patterns recorded under 0-20 kOe field. The nanosized ($\sim$20 nm) particles of BFO are anchored onto the graphene sheets of RGO via Fe-C bonds with concomitant rise in covalency in the Fe-O bonds. The field-dependent competition between the positive and negative magnetoelectric coupling arising from magnetostriction due to, respectively, interface and bulk magnetization appears to be giving rise to the observed nonmonotonic field dependence of polarization. The emergence of Fe-C bonds and consequent change in the magnetic and electronic structure of the interface region has influenced the coupling between ferroelectric and magnetic properties remarkably and thus creates a new way of tuning the magnetoelectric properties via reconstruction of interfaces in nanocomposites or heterostructures of graphene/single-phase-multiferroic systems. △ Less

Submitted 13 October, 2021; originally announced October 2021.

Comments: 10 pages, 12 figures, comments are welcome

Journal ref: Phys. Status Solidi RRL 16, 2200077 (2022)

Qurzon: A Prototype for a Divide and Conquer Based Quantum Compiler

Authors: Turbasu Chatterjee, Arnav Das, Shah Ishmam Mohtashim, Amit Saha, Amlan Chakrabarti

Abstract: When working with algorithms on quantum devices, quantum memory becomes a crucial bottleneck due to low qubit count in NISQ-era devices. In this context, the concept of `divide and compute', wherein a quantum circuit is broken into several subcircuits and executed separately, while stitching the results of the circuits via classical post-processing, becomes a viable option, especially in NISQ-era… ▽ More When working with algorithms on quantum devices, quantum memory becomes a crucial bottleneck due to low qubit count in NISQ-era devices. In this context, the concept of `divide and compute', wherein a quantum circuit is broken into several subcircuits and executed separately, while stitching the results of the circuits via classical post-processing, becomes a viable option, especially in NISQ-era devices. This paper introduces \textbf{Qurzon}, a proposed novel quantum compiler that incorporates the marriage of techniques of divide and compute with the state-of-the-art algorithms of optimal qubit placement for executing on real quantum devices. A scheduling algorithm is also introduced within the compiler that can explore the power of distributed quantum computing while paving the way for quantum parallelism for large algorithms. Several benchmark circuits have been executed using the compiler, thereby demonstrating the power of the divide and compute when working with real NISQ-era quantum devices. △ Less

Submitted 24 November, 2021; v1 submitted 15 September, 2021; originally announced September 2021.

Journal ref: SN COMPUT. SCI. 3, 323 (2022)

A near-term quantum simulation of the transverse field Ising model hints at Glassy Dynamics

Authors: Shah Ishmam Mohtashim, Arnav Das, Turbasu Chatterjee, Farhan Tanvir Chowdhury

Abstract: We demonstrate quantum circuit simulations of the transverse field Ising model with longitudinal fields, displaying salient features of glassy dynamics. The energy landscape and spin configurations of toy models are considered, using the Variational Quantum Eigensolver (VQE) to obtain the ground-state energies and corresponding eigenstates for a $6 \times 6$ Ising lattice using 36 qubits and a 1-d… ▽ More We demonstrate quantum circuit simulations of the transverse field Ising model with longitudinal fields, displaying salient features of glassy dynamics. The energy landscape and spin configurations of toy models are considered, using the Variational Quantum Eigensolver (VQE) to obtain the ground-state energies and corresponding eigenstates for a $6 \times 6$ Ising lattice using 36 qubits and a 1-dimensional Ising chain of length 25 using 25 qubits. The former showed disordered spin configurations for a specific mixture of values of the two fields. These insights mirror catalytic processes, where disorder within a catalyst can lead to inefficient reaction mechanisms. Results obtained from our proof-of-principle implementation make the case for kick-starting more concentrated efforts in harnessing existing quantum computational tools for computationally probing complex dynamical behaviour arising in quantum matter. Our aim is to leverage tools from quantum information processing to bring about a more nuanced understanding of the dynamics and structure of glassy systems, ultimately informing the development of novel materials and technology. △ Less

Submitted 21 April, 2025; v1 submitted 21 June, 2021; originally announced June 2021.

Comments: 12 pages, 5 figures, 1 table, accepted in special issue on "New Trends in Quantum Computing Technology" in EPJ ST

Journal ref: Eur. Phys. J. Spec. Top. 234, 6309-6322 (2025)

Detecting over/under-translation errors for determining adequacy in human translations

Authors: Prabhakar Gupta, Ridha Juneja, Anil Nelakanti, Tamojit Chatterjee

Abstract: We present a novel approach to detecting over and under translations (OT/UT) as part of adequacy error checks in translation evaluation. We do not restrict ourselves to machine translation (MT) outputs and specifically target applications with human generated translation pipeline. The goal of our system is to identify OT/UT errors from human translated video subtitles with high error recall. We ac… ▽ More We present a novel approach to detecting over and under translations (OT/UT) as part of adequacy error checks in translation evaluation. We do not restrict ourselves to machine translation (MT) outputs and specifically target applications with human generated translation pipeline. The goal of our system is to identify OT/UT errors from human translated video subtitles with high error recall. We achieve this without reference translations by learning a model on synthesized training data. We compare various classification networks that we trained on embeddings from pre-trained language model with our best hybrid network of GRU + CNN achieving 89.3% accuracy on high-quality human-annotated evaluation data in 8 languages. △ Less

Submitted 1 April, 2021; originally announced April 2021.

Comments: 6 pages, 5 tables

Study of parity sheaves arising from graded Lie algebra

Authors: Tamanna Chatterjee

Abstract: Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on… ▽ More Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. △ Less

Submitted 24 July, 2020; originally announced July 2020.

Comments: 39 pages. Comments are welcome!

MSC Class: 17B10; 20G05; 17B10

On partisan bias in redistricting: computational complexity meets the science of gerrymandering

Authors: Tanima Chatterjee, Bhaskar DasGupta

Abstract: The topic of this paper is "gerrymandering", namely the curse of deliberate creations of district maps with highly asymmetric electoral outcomes to disenfranchise voters, and it has a long legal history. Measuring and eliminating gerrymandering has enormous implications to sustain the backbone of democratic principles of a society. Although there is no dearth of legal briefs involving gerrymanderi… ▽ More The topic of this paper is "gerrymandering", namely the curse of deliberate creations of district maps with highly asymmetric electoral outcomes to disenfranchise voters, and it has a long legal history. Measuring and eliminating gerrymandering has enormous implications to sustain the backbone of democratic principles of a society. Although there is no dearth of legal briefs involving gerrymandering over many years, it is only more recently that mathematicians and applied computational researchers have started to investigate this topic. However, it has received relatively little attention so far from the computational complexity researchers dealing with theoretical analysis of computational complexity issues, such as computational hardness, approximability issues, etc. There could be many reasons for this, such as descriptions of these problem non-CS non-math (often legal or political) journals that theoretical CS (TCS) people usually do not follow, or the lack of coverage of these topics in TCS publication venues. One of our modest goals in writing this article is to improve upon this situation by stimulating further interactions between the gerrymandering and TCS researchers. To this effect, our main contributions are twofold: (1) we provide formalization of several models, related concepts, and corresponding problem statements using TCS frameworks from the descriptions of these problems as available in existing non-TCS (perhaps legal) venues, and (2) we also provide computational complexity analysis of some versions of these problems, leaving other versions for future research. The goal of writing this article is not to have the final word on gerrymandering, but to introduce a series of concepts, models and problems to the TCS community and to show that science of gerrymandering involves an intriguing set of partitioning problems involving geometric and combinatorial optimization. △ Less

Submitted 3 October, 2019; originally announced October 2019.

Comments: Disclaimer: The authors were not supported, financially or otherwise, by any political party. The research results reported in this paper are purely scientific and reported as they are without any regard to which political party they may be of help (if at all)

MSC Class: 68Q17; 68Q25; 68W20; 68W25; 68W40; 90C59 ACM Class: F.2.2; G.2.3; G.3; G.4; J.1; K.4

Journal ref: Journal of Combinatorial Optimization, 40(2), 512-546, 2020

Shifted Euler constants and a generalization of Euler-Stieltjes constants

Authors: Tapas Chatterjee, Suraj Singh Khurana

Abstract: The purpose of this article is twofold. First, we introduce the constants $ζ_k(α,r,q)$ where $α\in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $γ(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence condi… ▽ More The purpose of this article is twofold. First, we introduce the constants $ζ_k(α,r,q)$ where $α\in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $γ(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants $γ_k(χ)$ for a principal character $χ.$ In particular, we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant $γ_1(r/q)$ which was given by Blagouchine in 2015. △ Less

Submitted 11 July, 2019; originally announced July 2019.

Comments: 27 pages

MSC Class: 11M06; 11M99; 11Y60; 11M35; 11K65

Journal ref: Journal of Number Theory, 2019

Linear Independence of Harmonic Numbers over the field of Algebraic Numbers

Authors: Tapas Chatterjee, Sonika Dhillon

Abstract: Let $H_n =\sum\limits_{k=1}^n \frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the transcendental nature of $H_r$ for rational $r$. For some special values of $q>1,$ we give an upper bound for the number of linearly independent harmonic numbers… ▽ More Let $H_n =\sum\limits_{k=1}^n \frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the transcendental nature of $H_r$ for rational $r$. For some special values of $q>1,$ we give an upper bound for the number of linearly independent harmonic numbers $H_{a/q}$ with $ 1 \leq a \leq q$ over the field of algebraic numbers. Also, for any finite set of odd primes $J$ with $|J|=n,$ define $$W_J=\overline{\mathbb{Q}}-\text {span of } \{ H_1, \ H_{a_{j_i}/q_i} | \ 1 \leq a_{j_i} \leq q_i -1, \ 1 \leq j_i \leq q_i-1, \ \ \forall q_i \in J\}.$$ Finally, we show that $$\text{ dim }_{\overline{\mathbb{Q}}} ~W_J=\sum\limits_{\substack{i=1 \\ q_i \in J}}^n \frac{φ(q_i )}{2} + 2.$$ △ Less

Submitted 23 October, 2018; originally announced October 2018.

Comments: To appear in the Ramanujan Journal

MSC Class: 11J81; 11J86 (Primary); 11J91 (secondary)

A vanishing criterion for Dirichlet series with periodic coefficients

Authors: Tapas Chatterjee, M. Ram Murty, Siddhi Pathak

Abstract: We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the… ▽ More We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization. △ Less

Submitted 6 September, 2018; originally announced September 2018.

MSC Class: 11M06; 11M20

Journal ref: Contemp. Math., 701, 69--80, Amer. Math. Soc., Providence, RI, 2018

Alleviating partisan gerrymandering: can math and computers help to eliminate wasted votes?

Authors: Tanima Chatterjee, Bhaskar DasGupta, Laura Palmieri, Zainab Al-Qurashi, Anastasios Sidiropoulos

Abstract: Partisan gerrymandering is a major cause for voter disenfranchisement in United States. However, convincing US courts to adopt specific measures to quantify gerrymandering has been of limited success to date. Recently, Stephanopoulos and McGhee introduced a new and precise measure of partisan gerrymandering via the so-called "efficiency gap" that computes the absolutes difference of wasted votes b… ▽ More Partisan gerrymandering is a major cause for voter disenfranchisement in United States. However, convincing US courts to adopt specific measures to quantify gerrymandering has been of limited success to date. Recently, Stephanopoulos and McGhee introduced a new and precise measure of partisan gerrymandering via the so-called "efficiency gap" that computes the absolutes difference of wasted votes between two political parties in a two-party system. Quite importantly from a legal point of view, this measure was found legally convincing enough in a US appeals court in a case that claims that the legislative map of the state of Wisconsin was gerrymandered; the case is now pending in US Supreme Court. In this article, we show the following: (a) We provide interesting mathematical and computational complexity properties of the problem of minimizing the efficiency gap measure. To the best of our knowledge, these are the first non-trivial theoretical and algorithmic analyses of this measure of gerrymandering. (b) We provide a simple and fast algorithm that can "un-gerrymander" the district maps for the states of Texas, Virginia, Wisconsin and Pennsylvania by bring their efficiency gaps to acceptable levels from the current unacceptable levels. Our work thus shows that, notwithstanding the general worst case approximation hardness of the efficiency gap measure as shown by us,finding district maps with acceptable levels of efficiency gaps is a computationally tractable problem from a practical point of view. Based on these empirical results, we also provide some interesting insights into three practical issues related the efficiency gap measure. We believe that, should the US Supreme Court uphold the decision of lower courts, our research work and software will provide a crucial supporting hand to remove partisan gerrymandering. △ Less

Submitted 27 April, 2018; originally announced April 2018.

MSC Class: 68Q17; 68Q25; 68W20; 68W25; 68W40; 90C59 ACM Class: F.2.2; G.2.3; G.3; G.4; J.1; K.4

On the Computational Complexities of Three Privacy Measures for Large Networks Under Active Attack

Authors: Tanima Chatterjee, Bhaskar DasGupta, Nasim Mobasheri, Venkatkumar Srinivasan, Ismael G. Yero

Abstract: With the arrival of modern internet era, large public networks of various types have come to existence to benefit the society as a whole and several research areas such as sociology, economics and geography in particular. However, the societal and research benefits of these networks have also given rise to potentially significant privacy issues in the sense that malicious entities may violate the… ▽ More With the arrival of modern internet era, large public networks of various types have come to existence to benefit the society as a whole and several research areas such as sociology, economics and geography in particular. However, the societal and research benefits of these networks have also given rise to potentially significant privacy issues in the sense that malicious entities may violate the privacy of the users of such a network by analyzing the network and deliberately using such privacy violations for deleterious purposes. Such considerations have given rise to a new active research area that deals with the quantification of privacy of users in large networks and the corresponding investigation of computational complexity issues of computing such quantified privacy measures. In this paper, we formalize three such privacy measures for large networks and provide non-trivial theoretical computational complexity results for computing these measures. Our results show the first two measures can be computed efficiently, whereas the third measure is provably hard to compute within a logarithmic approximation factor. Furthermore, we also provide computational complexity results for the case when the privacy requirement of the network is severely restricted, including an efficient logarithmic approximation. △ Less

Submitted 5 July, 2016; originally announced July 2016.

Comments: 21 pages, 3 figures

MSC Class: 68Q17; 68Q25; 68R10; 05C85; 68W25; 68W40 ACM Class: E.1; F.2.2; G.2.1; G.2.2; G.2.3; G.4; I.1.2

A number field extension of a question of Milnor

Authors: Tapas Chatterjee, Sanoli Gun, Purusottam Rath

Abstract: Milnor formulated a conjecture about rational linear independence of some special Hurwitz zeta values. The second and third authors along with Ram Murty studied this conjecture and suggested an extension of Milnor's conjecture. In this note, we investigate the number field generalisation of this extended Milnor conjecture. We indicate the motivation for considering this number field case by noting… ▽ More Milnor formulated a conjecture about rational linear independence of some special Hurwitz zeta values. The second and third authors along with Ram Murty studied this conjecture and suggested an extension of Milnor's conjecture. In this note, we investigate the number field generalisation of this extended Milnor conjecture. We indicate the motivation for considering this number field case by noting that such a phenomenon is true in an analogous context. We also study some new spaces related to normalised Hurwitz zeta values. △ Less

Submitted 10 January, 2016; originally announced January 2016.

MSC Class: 11M35; 11R18; 11R21; 11J86; 11R32

Journal ref: Contemporary Mathematics, AMS,655, (2015), 15--26

On a conjecture of Erdös and certain Dirichlet series

Authors: Tapas Chatterjee, M. Ram Murty

Abstract: Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$). Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$). △ Less

Submitted 17 January, 2015; originally announced January 2015.

Comments: To appear in the Pacific Journal of Mathematics

MSC Class: 11M06; 11M41; 11M20

Journal ref: Pacific Journal of Mathematics 275-1 (2015), 103--113

On the zeros of generalized Hurwitz zeta functions

Authors: Tapas Chatterjee, Sanoli Gun

Abstract: In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function. In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function. △ Less

Submitted 31 July, 2014; originally announced July 2014.

Comments: To appear in Journal of Number Theory

Non-vanishing of Dirichlet series with periodic coefficients

Authors: Tapas Chatterjee, M. Ram Murty

Abstract: For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at $s=1$, where it has a simple pole with residue $$ρ:= q^{-1}\sum_{1\leq a\leq q} f(a).$$ Thus, the function is analytic at $s=1$ when $ρ=0$ and in this case, we stu… ▽ More For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at $s=1$, where it has a simple pole with residue $$ρ:= q^{-1}\sum_{1\leq a\leq q} f(a).$$ Thus, the function is analytic at $s=1$ when $ρ=0$ and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet $L$-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of $L(1,f)$ as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of $L(1,f)$. △ Less

Submitted 27 May, 2014; originally announced May 2014.

Comments: 22 pages

MSC Class: 11M06; 11M20

VHDL Modeling of Intrusion Detection & Prevention System (IDPS) A Neural Network Approach

Authors: Tanusree Chatterjee, Abhishek Bhattacharya

Abstract: The rapid development and expansion of World Wide Web and network systems have changed the computing world in the last decade and also equipped the intruders and hackers with new facilities for their destructive purposes. The cost of temporary or permanent damages caused by unauthorized access of the intruders to computer systems has urged different organizations to increasingly implement various… ▽ More The rapid development and expansion of World Wide Web and network systems have changed the computing world in the last decade and also equipped the intruders and hackers with new facilities for their destructive purposes. The cost of temporary or permanent damages caused by unauthorized access of the intruders to computer systems has urged different organizations to increasingly implement various systems to monitor data flow in their network. The systems are generally known as Intrusion Detection System (IDS).Our objective is to implement an artificial network approach to the design of intrusion detection and prevention system and finally convert the designed model to a VHDL (Very High Speed Integrated Circuit Hardware Descriptive Language) code. This feature enables the system to suggest proper actions against possible attacks. The promising results of the present study show the potential applicability of ANNs for developing practical IDSs. △ Less

Submitted 21 February, 2014; originally announced February 2014.

An Estimation Method of Measuring Image Quality for Compressed Images of Human Face

Authors: Abhishek Bhattacharya, Tanusree Chatterjee

Abstract: Nowadays digital image compression and decompression techniques are very much important. So our aim is to calculate the quality of face and other regions of the compressed image with respect to the original image. Image segmentation is typically used to locate objects and boundaries (lines, curves etc.)in images. After segmentation the image is changed into something which is more meaningful to an… ▽ More Nowadays digital image compression and decompression techniques are very much important. So our aim is to calculate the quality of face and other regions of the compressed image with respect to the original image. Image segmentation is typically used to locate objects and boundaries (lines, curves etc.)in images. After segmentation the image is changed into something which is more meaningful to analyze. Using Universal Image Quality Index(Q),Structural Similarity Index(SSIM) and Gradient-based Structural Similarity Index(G-SSIM) it can be shown that face region is less compressed than any other region of the image. △ Less

Submitted 6 February, 2014; originally announced February 2014.

Comments: 6 pages

The Digamma function, Euler-Lehmer constants and their $p$-adic counterparts

Authors: Tapas Chatterjee, Sanoli Gun

Abstract: The goal of this article is twofold. We first extend a result of Murty and Saradha \cite{MS} related to the digamma function at rational arguments. Further, we extend another result of the same authors \cite{MS1} about the nature of $p$-adic Euler-Lehmer constants. The goal of this article is twofold. We first extend a result of Murty and Saradha \cite{MS} related to the digamma function at rational arguments. Further, we extend another result of the same authors \cite{MS1} about the nature of $p$-adic Euler-Lehmer constants. △ Less

Submitted 29 August, 2013; originally announced August 2013.

Comments: 13 pages

Journal ref: Acta Arithmetica 162 (2014), 197-208

The Strong Chowla-Milnor spaces and a conjecture of Gun, Murty and Rath

Authors: Tapas Chatterjee

Abstract: In a recent work, Gun, Murty and Rath formulated the Strong Chowla-Milnor conjecture and defined the Strong Chowla-Milnor space. In this paper, we prove a non-trivial lower bound for the dimension of these spaces. We also obtain a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both $ζ(k)$ and $ζ(k)/ π^k$ for… ▽ More In a recent work, Gun, Murty and Rath formulated the Strong Chowla-Milnor conjecture and defined the Strong Chowla-Milnor space. In this paper, we prove a non-trivial lower bound for the dimension of these spaces. We also obtain a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both $ζ(k)$ and $ζ(k)/ π^k$ for all odd positive integers $k>1$. Following Gun, Murty and Rath, we define generalized Zagier spaces $V_p(K)$ for multiple zeta values over a number field $K$. We prove that the dimension of $V_{4d+2}(K)$ for $d\geq 1$, is at least 2, assuming a conjecture of Gun, Murty and Rath. △ Less

Submitted 29 August, 2013; originally announced August 2013.

Comments: 15 pages

Journal ref: Int. J. Number Theory, (8) (5),(2012), 1301-1314