Showing 1–38 of 38 results for author: Zaman, A

Numerical Computations Concerning Landau-Siegel Zeros

Authors: Rick F. Lu, Asif Zaman, Haonan Zhao

Abstract: We computationally verify that if $L(s,χ)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(σ,χ) \neq 0$ for real $σ\ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins (2004), Platt (2016), and Languasco (2023) by a factor between 66 and 25,000. Our new algorithm draws from zero-free region arguments. We computationally verify that if $L(s,χ)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(σ,χ) \neq 0$ for real $σ\ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins (2004), Platt (2016), and Languasco (2023) by a factor between 66 and 25,000. Our new algorithm draws from zero-free region arguments. △ Less

Submitted 3 February, 2026; originally announced February 2026.

Comments: 20 pages, 2 figures, and 1 table

MSC Class: 11M20 (Primary); 11M06 (Secondary)

The least prime with a given cycle type

Authors: Peter J. Cho, Robert J. Lemke Oliver, Asif Zaman

Abstract: Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^α$ for some constan… ▽ More Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^α$ for some constant $α= α(G,C)$. There is a rich literature establishing unconditional admissible values for $α$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $α$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $n\geq 2$ and any conjugacy class $C \subset S_n$, one may take $α(S_n,C) = c_1 \exp(-c_2n)$. Our approach reduces the core problem to a question in character theory. △ Less

Submitted 31 December, 2025; originally announced December 2025.

Comments: 44 pages, 9 tables

MSC Class: 11R44; 11R47

Central limit theorems for random multiplicative functions over function fields

Authors: Declan Hoban, Jibran Iqbal Shah, Nadya-Catherine Ismail, William Verreault, Asif Zaman

Abstract: We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in s… ▽ More We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals. △ Less

Submitted 7 December, 2025; v1 submitted 28 November, 2025; originally announced November 2025.

Comments: 31 pages; added a few references and Remark 2.2.1

Journal ref: Res. Math. Sci. 13 (2026), no. 58

Effective Brauer-Siegel theorems for Artin $L$-functions

Authors: Peter J. Cho, Robert J. Lemke Oliver, Asif Zaman

Abstract: Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-fu… ▽ More Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions. △ Less

Submitted 26 June, 2026; v1 submitted 2 October, 2025; originally announced October 2025.

Comments: 22 pages; fixed various typos

MSC Class: 11M20; 11M41; 11R42

A guide to Tauberian theorems for arithmetic applications

Authors: Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, Asif Zaman

Abstract: A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number th… ▽ More A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened. △ Less

Submitted 2 July, 2026; v1 submitted 22 April, 2025; originally announced April 2025.

Comments: 83 pages; v4 corrects only a few small typos in v3, found when checking page proofs

Improving the trivial bound for $\ell$-torsion in class groups

Authors: Robert J. Lemke Oliver, Asif Zaman

Abstract: For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit… ▽ More For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\ell$. △ Less

Submitted 5 February, 2025; originally announced February 2025.

Comments: 11 pages

MSC Class: 11R29; 11R42

Distributed Offloading in Multi-Access Edge Computing Systems: A Mean-Field Perspective

Authors: Shubham Aggarwal, Muhammad Aneeq uz Zaman, Melih Bastopcu, Sennur Ulukus, Tamer Başar

Abstract: Multi-access edge computing (MEC) technology is a promising solution to assist power-constrained IoT devices by providing additional computing resources for time-sensitive tasks. In this paper, we consider the problem of optimal task offloading in MEC systems with due consideration of the timeliness and scalability issues under two scenarios of equitable and priority access to the edge server (ES)… ▽ More Multi-access edge computing (MEC) technology is a promising solution to assist power-constrained IoT devices by providing additional computing resources for time-sensitive tasks. In this paper, we consider the problem of optimal task offloading in MEC systems with due consideration of the timeliness and scalability issues under two scenarios of equitable and priority access to the edge server (ES). In the first scenario, we consider a MEC system consisting of $N$ devices assisted by one ES, where the devices can split task execution between a local processor and the ES, with equitable access to the ES. In the second scenario, we consider a MEC system consisting of one primary user, $N$ secondary users and one ES. The primary user has priority access to the ES while the secondary users have equitable access to the ES amongst themselves. In both scenarios, due to the power consumption associated with utilizing the local resource and task offloading, the devices must optimize their actions. Additionally, since the ES is a shared resource, other users' offloading activity serves to increase latency incurred by each user. We thus model both scenarios using a non-cooperative game framework. However, the presence of a large number of users makes it nearly impossible to compute the equilibrium offloading policies for each user, which would require a significant information exchange overhead between users. Thus, to alleviate such scalability issues, we invoke the paradigm of mean-field games to compute approximate Nash equilibrium policies for each user using their local information, and further study the trade-offs between increasing information freshness and reducing power consumption for each user. Using numerical evaluations, we show that our approach can recover the offloading trends displayed under centralized solutions, and provide additional insights into the results obtained. △ Less

Submitted 30 January, 2025; originally announced January 2025.

Comments: Submitted to IEEE for possible publication

Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions

Authors: Kübra Benli, Shivani Goel, Henry Twiss, Asif Zaman

Abstract: Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman. Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman. △ Less

Submitted 8 January, 2026; v1 submitted 8 October, 2024; originally announced October 2024.

MSC Class: 11M06; 11M20; 11N36

Moments of random multiplicative functions over function fields

Authors: Maximilian C. E. Hofmann, Annemily Hoganson, Siddarth Menon, William Verreault, Asif Zaman

Abstract: Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic ar… ▽ More Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment. △ Less

Submitted 15 August, 2024; originally announced August 2024.

Comments: 32 pages

Journal ref: Math. Proc. Camb. Phil. Soc. 179 (2025) 785-819

Semantic Communication in Multi-team Dynamic Games: A Mean Field Perspective

Authors: Shubham Aggarwal, Muhammad Aneeq uz Zaman, Melih Bastopcu, Tamer Başar

Abstract: Coordinating communication and control is a key component in the stability and performance of networked multi-agent systems. While single user networked control systems have gained a lot of attention within this domain, in this work, we address the more challenging problem of large population multi-team dynamic games. In particular, each team constitutes two decision makers (namely, the sensor and… ▽ More Coordinating communication and control is a key component in the stability and performance of networked multi-agent systems. While single user networked control systems have gained a lot of attention within this domain, in this work, we address the more challenging problem of large population multi-team dynamic games. In particular, each team constitutes two decision makers (namely, the sensor and the controller) who coordinate over a shared network to control a dynamically evolving state of interest under costs on both actuation and sensing/communication. Due to the shared nature of the wireless channel, the overall cost of each team depends on other teams' policies, thereby leading to a noncooperative game setup. Due to the presence of a large number of teams, we compute approximate decentralized Nash equilibrium policies for each team using the paradigm of (extended) mean-field games, which is governed by (1) the mean traffic flowing over the channel, and (2) the value of information at the sensor, which highlights the semantic nature of the ensuing communication. In the process, we compute optimal controller policies and approximately optimal sensor policies for each representative team of the mean-field system to alleviate the problem of general non-contractivity of the mean-field fixed point operator associated with the finite cardinality of the sensor action space. Consequently, we also prove the $ε$--Nash property of the mean-field equilibrium solution which essentially characterizes how well the solution derived using mean-field analysis performs on the finite-team system. Finally, we provide extensive numerical simulations, which corroborate the theoretical findings and lead to additional insights on the properties of the results presented. △ Less

Submitted 24 June, 2025; v1 submitted 8 July, 2024; originally announced July 2024.

Comments: Submitted to IEEE for possible publication

Large Population Games on Constrained Unreliable Networks

Authors: Shubham Aggarwal, Muhammad Aneeq uz Zaman, Melih Bastopcu, Tamer Başar

Abstract: This paper studies an $N$--agent cost-coupled game where the agents are connected via an unreliable capacity constrained network. Each agent receives state information over that network which loses packets with probability $p$. A Base station (BS) actively schedules agent communications over the network by minimizing a weighted Age of Information (WAoI) based cost function under a capacity limit… ▽ More This paper studies an $N$--agent cost-coupled game where the agents are connected via an unreliable capacity constrained network. Each agent receives state information over that network which loses packets with probability $p$. A Base station (BS) actively schedules agent communications over the network by minimizing a weighted Age of Information (WAoI) based cost function under a capacity limit $\mathcal{C} < N$ on the number of transmission attempts at each instant. Under a standard information structure, we show that the problem can be decoupled into a scheduling problem for the BS and a game problem for the $N$ agents. Since the scheduling problem is an NP hard combinatorics problem, we propose an approximately optimal solution which approaches the optimal solution as $N \rightarrow \infty$. In the process, we also provide some insights on the case without channel erasure. Next, to solve the large population game problem, we use the mean-field game framework to compute an approximate decentralized Nash equilibrium. Finally, we validate the theoretical results using a numerical example. △ Less

Submitted 16 March, 2023; originally announced March 2023.

Comments: Submitted to IEEE for possible publication

Weighted Age of Information based Scheduling for Large Population Games on Networks

Authors: Shubham Aggarwal, Muhammad Aneeq uz Zaman, Melih Bastopcu, Tamer Başar

Abstract: In this paper, we consider a discrete-time multi-agent system involving $N$ cost-coupled networked rational agents solving a consensus problem and a central Base Station (BS), scheduling agent communications over a network. Due to a hard bandwidth constraint on the number of transmissions through the network, at most $R_d < N$ agents can concurrently access their state information through the netw… ▽ More In this paper, we consider a discrete-time multi-agent system involving $N$ cost-coupled networked rational agents solving a consensus problem and a central Base Station (BS), scheduling agent communications over a network. Due to a hard bandwidth constraint on the number of transmissions through the network, at most $R_d < N$ agents can concurrently access their state information through the network. Under standard assumptions on the information structure of the agents and the BS, we first show that the control actions of the agents are free of any dual effect, allowing for separation between estimation and control problems at each agent. Next, we propose a weighted age of information (WAoI) metric for the scheduling problem of the BS, where the weights depend on the estimation error of the agents. The BS aims to find the optimum scheduling policy that minimizes the WAoI, subject to the hard bandwidth constraint. Since this problem is NP hard, we first relax the hard constraint to a soft update rate constraint, and then compute an optimal policy for the relaxed problem by reformulating it into a Markov Decision Process (MDP). This then inspires a sub-optimal policy for the bandwidth constrained problem, which is shown to approach the optimal policy as $N \rightarrow \infty$. Next, we solve the consensus problem using the mean-field game framework wherein we first design decentralized control policies for a limiting case of the $N$-agent system (as $N \rightarrow \infty$). By explicitly constructing the mean-field system, we prove the existence and uniqueness of the mean-field equilibrium. Consequently, we show that the obtained equilibrium policies constitute an $ε$-Nash equilibrium for the finite agent system. Finally, we validate the performance of both the scheduling and the control policies through numerical simulations. △ Less

Submitted 26 December, 2022; v1 submitted 26 September, 2022; originally announced September 2022.

Comments: This work has been submitted to IEEE for possible publication

An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes

Authors: Jesse Thorner, Asif Zaman

Abstract: We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $σ= 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of Sárközy. We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $σ= 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of Sárközy. △ Less

Submitted 4 August, 2023; v1 submitted 23 August, 2022; originally announced August 2022.

Comments: 24 pages. v4: Incorporates referee comments

Journal ref: Forum Math. 36 (2024), no. 4, 1059-1080

Observer-Based Consensus of Nonlinear Positive Multi-Agent Systems with Saturated Control Input

Authors: Amirreza Zaman, Wolfgang Birk, Khalid Tourkey Atta

Abstract: This paper presents the distributed pinning consensus solution for nonlinear positive multi-agent systems with nonlinear control input by applying observer-based control protocols. The network topology is considered as a directed and fully connected structure. By considering sector input nonlinearities and various forms of topologies, two kinds of state observers involving standard observer and di… ▽ More This paper presents the distributed pinning consensus solution for nonlinear positive multi-agent systems with nonlinear control input by applying observer-based control protocols. The network topology is considered as a directed and fully connected structure. By considering sector input nonlinearities and various forms of topologies, two kinds of state observers involving standard observer and distributed pinning observer are presented for each regarded nonlinear agent by applying a novel analysis directly dealing with the nonlinear input and nonlinear system dynamics. The measured local output detail outlines the first observer, and the other observer is achieved via the corresponding output detail of its adjacent agents. Based on further observed state details, a distributed pinning observer-based strategy is derived for the leader-follower non-negative global consensus of the nonlinear positive multi-agent system. Additionally, two multi-step algorithms are proposed to set up the observer gains and each protocol criterion. Performance evaluations are provided to confirm the proposed control method and illustrate the effectiveness of the derived non-negative consensus observer-based protocols. △ Less

Submitted 12 May, 2022; originally announced May 2022.

Comments: 27 pages, 6 figures, Pre-print submitted to Elsevier

Distributed Nonparametric Estimation under Communication Constraints

Authors: Azeem Zaman, Botond Szabó

Abstract: In the era of big data, it is necessary to split extremely large data sets across multiple computing nodes and construct estimators using the distributed data. When designing distributed estimators, it is desirable to minimize the amount of communication across the network because transmission between computers is slow in comparison to computations in a single computer. Our work provides a general… ▽ More In the era of big data, it is necessary to split extremely large data sets across multiple computing nodes and construct estimators using the distributed data. When designing distributed estimators, it is desirable to minimize the amount of communication across the network because transmission between computers is slow in comparison to computations in a single computer. Our work provides a general framework for understanding the behavior of distributed estimation under communication constraints for nonparametric problems. We provide results for a broad class of models, moving beyond the Gaussian framework that dominates the literature. As concrete examples we derive minimax lower and matching upper bounds in the distributed regression, density estimation, classification, Poisson regression and volatility estimation models under communication constraints. To assist with this, we provide sufficient conditions that can be easily verified in all of our examples. △ Less

Submitted 21 April, 2022; originally announced April 2022.

Linear Quadratic Mean-Field Games with Communication Constraints

Authors: Shubham Aggarwal, Muhammad Aneeq uz Zaman, Tamer Başar

Abstract: In this paper, we study a large population game with heterogeneous dynamics and cost functions solving a consensus problem. Moreover, the agents have communication constraints which appear as: (1) an Additive-White Gaussian Noise (AWGN) channel, and (2) asynchronous data transmission via a fixed scheduling policy. Since the complexity of solving the game increases with the number of agents, we use… ▽ More In this paper, we study a large population game with heterogeneous dynamics and cost functions solving a consensus problem. Moreover, the agents have communication constraints which appear as: (1) an Additive-White Gaussian Noise (AWGN) channel, and (2) asynchronous data transmission via a fixed scheduling policy. Since the complexity of solving the game increases with the number of agents, we use the Mean-Field Game paradigm to solve it. Under standard assumptions on the information structure of the agents, we prove that the control of the agent in the MFG setting is free of the dual effect. This allows us to obtain an equilibrium control policy for the generic agent, which is a function of only the local observation of the agent. Furthermore, the equilibrium mean-field trajectory is shown to follow linear dynamics, hence making it computable. We show that in the finite population game, the equilibrium control policy prescribed by the MFG analysis constitutes an $ε$-Nash equilibrium, where $ε$ tends to zero as the number of agents goes to infinity. The paper is concluded with simulations demonstrating the performance of the equilibrium control policy. △ Less

Submitted 25 August, 2022; v1 submitted 10 March, 2022; originally announced March 2022.

Comments: Accepted in American Control Conference 2022

A conjectural asymptotic formula for multiplicative chaos in number theory

Authors: Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng

Abstract: We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random m… ▽ More We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when $A(N)$ is defined using standard real Gaussians or uniform $\pm 1$ random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure. △ Less

Submitted 25 August, 2021; originally announced August 2021.

Comments: 18 pages

MSC Class: 60-08 (Primary); 60-11; 60F99; 11K65 (Secondary)

Journal ref: Res. Number Theory 8, 35 (2022)

Refinements to the prime number theorem for arithmetic progressions

Authors: Jesse Thorner, Asif Zaman

Abstract: We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-δ}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region, a log-free zero density estimate, and the D… ▽ More We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-δ}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region, a log-free zero density estimate, and the Deuring-Heilbronn zero repulsion phenomenon. Improvements exist when the modulus is sufficiently powerful. △ Less

Submitted 21 September, 2021; v1 submitted 24 August, 2021; originally announced August 2021.

Comments: 11 pages. Theorems 1.1 and 2.1 improved

MSC Class: 11N05 (Primary); 11N13; 11M06 (Secondary)

Journal ref: Math. Z. 306 (2024), no. 3, Paper No. 54

Sums of random multiplicative functions over function fields with few irreducible factors

Authors: Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng

Abstract: We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers. We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers. △ Less

Submitted 28 January, 2022; v1 submitted 18 August, 2021; originally announced August 2021.

Comments: 10 pages. Simplification of the proof of Lemma 5 and typos corrected, one reference added

MSC Class: 11K65 (Primary) 60F05; 60G50 (Secondary)

Journal ref: Math. Proc. Camb. Phil. Soc. (2022), 1-12

A model problem for multiplicative chaos in number theory

Authors: Kannan Soundararajan, Asif Zaman

Abstract: Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture o… ▽ More Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area. △ Less

Submitted 16 August, 2021; originally announced August 2021.

Comments: 25 pages

Journal ref: Enseign. Math. 68 (2022), no. 3, 307-340

An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions

Authors: Robert J. Lemke Oliver, Jesse Thorner, Asif Zaman

Abstract: Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the symmetric group $S_n$ or any transitive group of prime degree, then we unconditionally prove that for all $K\in\mathfrak{F}_k^G(Q)$ with at most $O_ε(Q^ε)$ exce… ▽ More Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the symmetric group $S_n$ or any transitive group of prime degree, then we unconditionally prove that for all $K\in\mathfrak{F}_k^G(Q)$ with at most $O_ε(Q^ε)$ exceptions, the $L$-functions associated to the faithful Artin representations of $\mathrm{Gal}(K/k)$ have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: 1) there exist infinitely many degree $n$ $S_n$-fields over $\mathbb{Q}$ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; 2) for a prime $p$, the periodic torus orbits attached to the ideal classes of almost all totally real degree $p$ fields $F$ over $\mathbb{Q}$ equidistribute on $\mathrm{PGL}_p(\mathbb{Z})\backslash\mathrm{PGL}_p(\mathbb{R})$ with respect to Haar measure; 3) for each $\ell\geq 2$, the $\ell$-torsion subgroups of the ideal class groups of almost all degree $p$ fields over $k$ (resp. almost all degree $n$ $S_n$-fields over $k$) are as small as GRH implies; and 4) an effective variant of the Chebotarev density theorem holds for almost all fields in such families. △ Less

Submitted 31 May, 2021; v1 submitted 28 December, 2020; originally announced December 2020.

Comments: v3: Reworked the first three sections

Journal ref: Invent. Math. 235 (2024), no. 3, 893-971

Multi-agent Planning for thermalling gliders using multi level graph-search

Authors: Muhammad Aneeq uz Zaman, Aamer Iqbal Bhatti

Abstract: This paper solves a path planning problem for a group of gliders. The gliders are tasked with visiting a set of interest points. The gliders have limited range but are able to increase their range by visiting special points called thermals. The problem addressed in this paper is of path planning for the gliders such that, the total number of interest points visited by the gliders is maximized. Thi… ▽ More This paper solves a path planning problem for a group of gliders. The gliders are tasked with visiting a set of interest points. The gliders have limited range but are able to increase their range by visiting special points called thermals. The problem addressed in this paper is of path planning for the gliders such that, the total number of interest points visited by the gliders is maximized. This is referred to as the multi-agent problem. The problem is solved by first decomposing it into several single-agent problems. In a single-agent problem a set of interest points are allocated to a single glider. This problem is solved by planning a path which maximizes the number of visited interest points from the allocated set. This is achieved through a uniform cost graph search, as shown in our earlier work. The multi-agent problem now consists of determining the best allocation (of interest points) for each glider. Two ways are presented of solving this problem, a brute force search approach as shown in earlier work and a Branch\&Bound type graph search. The Branch&Bound approach is the main contribution of the paper. This approach is proven to be optimal and shown to be faster than the brute force search using simulations. △ Less

Submitted 2 July, 2020; originally announced July 2020.

Security Analysis and Fault Detection Against Stealthy Replay Attacks

Authors: Amirreza Zaman, Behrouz Safarinejadian

Abstract: This paper investigates the security issue of the data replay attacks on the control systems. The attacker is assumed to interfere with the control system process in a steady-state case. The problem is presented as the standard way to attack, which is storing measurements and replay ing them in further times to the system. The controller is assumed to be the LQG controller. The main novelty in thi… ▽ More This paper investigates the security issue of the data replay attacks on the control systems. The attacker is assumed to interfere with the control system process in a steady-state case. The problem is presented as the standard way to attack, which is storing measurements and replay ing them in further times to the system. The controller is assumed to be the LQG controller. The main novelty in this paper can be stated as proposing a different attack detection criterion by using the K-L divergence method to cover more general control system problems with these attacks and with higher-order dynamics. Also, there exists a packet-dropout feature in transmitting the data as another contribution of the paper. Formulations and numerical simulations prove the effectiveness of the newly proposed attack detection procedure by having a quick response to occurred attacks. Although, in previous approaches, the trade-off between attack detection delay or LQG performance was significant, in this approach it is proved that the difference in this trade-off is not considered in early moments when the attack happens since the attack detection rate is rapid and thus, these attacks can be stopped with defense strategies in the first moments with the proposed attack detection criterion. △ Less

Submitted 28 August, 2020; v1 submitted 5 April, 2020; originally announced April 2020.

Comments: 12 pages, 5 figures, grammatical errors and typos corrected

A zero density estimate for Dedekind zeta functions

Authors: Jesse Thorner, Asif Zaman

Abstract: Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven progress towards the strong form of Artin's conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butter… ▽ More Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven progress towards the strong form of Artin's conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butterbaugh, and Wood on the average error in the Chebotarev density theorem and $\ell$-torsion in ideal class groups. △ Less

Submitted 5 January, 2022; v1 submitted 3 September, 2019; originally announced September 2019.

Comments: Considerably streamlined, small refinements to Theorems 1.1 and 1.2. 14 pages

Journal ref: Int. Math. Res. Not. IMRN 2023, no. 8, 6739-6761

An unconditional $\mathrm{GL}(n)$ large sieve

Authors: Jesse Thorner, Asif Zaman

Abstract: Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of $π\in\mathfrak{F}_n$, one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of $L$-functions… ▽ More Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of $π\in\mathfrak{F}_n$, one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of $L$-functions $L(s,π)$ associated to $π\in\mathfrak{F}_n$, which we make log-free. As an application of the zero density estimate, we prove a hybrid subconvexity bound for $L(\frac{1}{2},π)$ for a density one subset of $π\in\mathfrak{F}_n$. △ Less

Submitted 9 March, 2021; v1 submitted 18 June, 2019; originally announced June 2019.

Comments: 17 pages. Incorporates referee comments

Journal ref: Adv. Math. 378 (2021), 107529, 24 pp

Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

Authors: Farrell Brumley, Jesse Thorner, Asif Zaman

Abstract: Let $π$ and $π_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,π\timesπ_0)$, where $π$ varies in a given family and $π_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider appli… ▽ More Let $π$ and $π_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,π\timesπ_0)$, where $π$ varies in a given family and $π_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields. △ Less

Submitted 26 April, 2019; v1 submitted 17 April, 2018; originally announced April 2018.

Comments: 53 pages. Farrell Brumley added as coauthor. Added two appendices, one of which is authored by Colin Bushnell and Guy Henniart. Theorem 1.3 added, leads to cleaner versions of Theorems 1.2 and 1.4

Journal ref: J. Eur. Math. Soc. (JEMS) 24 (2022), no. 5, 1471-1541

A unified and improved Chebotarev density theorem

Authors: Jesse Thorner, Asif Zaman

Abstract: We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest uncond… ▽ More We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors. We also present a new application of our main result that exhibits considerable gains over earlier versions of the Chebotarev density theorem. If $f$ is a positive definite primitive binary quadratic form then we count lattice points $(u,v) \in \mathbb{Z}^2$ such that $f(u,v)$ is prime and $u, v$ have no prime factors $\leq z$ with uniformity in $z$ and the discriminant of $f$. △ Less

Submitted 26 October, 2018; v1 submitted 7 March, 2018; originally announced March 2018.

Comments: 26 pages; v3 intro revised and application added in Sections 6 and 7; typos corrected

MSC Class: 11R44

Journal ref: Alg. Number Th. 13 (2019) 1039-1068

Primes represented by positive definite binary quadratic forms

Authors: Asif Zaman

Abstract: Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $π_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $π_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of… ▽ More Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $π_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $π_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors. △ Less

Submitted 24 October, 2017; originally announced October 2017.

Comments: 28 pages

MSC Class: 11E16; 11N05

Journal ref: Q. J. Math. 69 (2018), no. 4, 1353-1386

The Density of Numbers Represented by Diagonal Forms of Large Degree

Authors: Brandon Hanson, Asif Zaman

Abstract: Let $s \geq 3$ be a fixed positive integer and $a_1,\dots,a_s \in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k \] decays rapidly with respect to $k$. Let $s \geq 3$ be a fixed positive integer and $a_1,\dots,a_s \in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k \] decays rapidly with respect to $k$. △ Less

Submitted 13 June, 2017; originally announced June 2017.

Comments: 7 pages

Journal ref: Mathematika 64 (2018) 542-550

The least unramified prime which does not split completely

Authors: Asif Zaman

Abstract: Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or split completely in $K$. We improve upon the previous best known general estimates due to X. Li when $F = \mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our… ▽ More Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or split completely in $K$. We improve upon the previous best known general estimates due to X. Li when $F = \mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F \neq \mathbb{Q}$. △ Less

Submitted 11 April, 2017; originally announced April 2017.

Comments: 13 pages

MSC Class: 11R44

Journal ref: Forum Math. 30 (2018), no. 3, 651-661

A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures

Authors: Jesse Thorner, Asif Zaman

Abstract: We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frob… ▽ More We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms. △ Less

Submitted 7 December, 2016; v1 submitted 29 June, 2016; originally announced June 2016.

Comments: 26 pages; v2 several typographical and minor errors are fixed, some of which had a small impact on the quality of the exponents in Thm 1.1, Thm 1.2, and Cor 1.3

MSC Class: 11R44

Journal ref: Int. Math. Res. Not. IMRN 2018, no. 16, 4991-5027

An explicit bound for the least prime ideal in the Chebotarev density theorem

Authors: Jesse Thorner, Asif Zaman

Abstract: We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove th… ▽ More We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms. △ Less

Submitted 17 August, 2016; v1 submitted 6 April, 2016; originally announced April 2016.

Comments: 45 pages. v1 subsumes arXiv:1510.08086 but adds too much new material to be considered a revised version; v2 exposition tightened, minor corrections, slight improvement in Theorem 1.1

Journal ref: Alg. Number Th. 11 (2017) 1135-1197

Explicit results on the distribution of zeros of Hecke $L$-functions

Authors: Jesse Thorner, Asif Zaman

Abstract: We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal… ▽ More We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem. △ Less

Submitted 27 October, 2015; originally announced October 2015.

Comments: 32 pages

MSC Class: 11R42; 11N36; 11R44

Journal ref: J. Number Theory 162 (2016), 312-275

Bounding the least prime ideal in the Chebotarev Density Theorem

Authors: Asif Zaman

Abstract: Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function. Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function. △ Less

Submitted 10 February, 2017; v1 submitted 2 August, 2015; originally announced August 2015.

Comments: 23 pages; v3 corrects typos and improves Theorem 1.3 & Corollary 1.4; accepted at Funct. Approx. Comment. Math. (2017)

Journal ref: Funct. Approx. Comment. Math. 57 (2017), no. 1, 115-142

On the least prime ideal and Siegel zeros

Authors: Asif Zaman

Abstract: Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $ψ$, possibly principal, with a Siegel zero $β$. For $\mathcal{C} \in \mathrm{Cl}(\mathfrak{q})$ satisfying $ψ(\mathcal{C}) = 1$, we establish an effective $K$-uniform Linnik-type bound with exp… ▽ More Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $ψ$, possibly principal, with a Siegel zero $β$. For $\mathcal{C} \in \mathrm{Cl}(\mathfrak{q})$ satisfying $ψ(\mathcal{C}) = 1$, we establish an effective $K$-uniform Linnik-type bound with explicit constants for the least norm of a prime ideal $\mathfrak{p} \in \mathcal{C}$. A special case of this result is related to rational primes represented by certain binary quadratic forms. △ Less

Submitted 4 June, 2015; originally announced June 2015.

Comments: 25 pages

MSC Class: 11R44

Journal ref: Int. J. Number Theory 12 (2016), no. 8, 2201-2229

Explicit estimates for the zeros of Hecke $L$-functions

Authors: Asif Zaman

Abstract: Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estima… ▽ More Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates. △ Less

Submitted 29 March, 2016; v1 submitted 19 February, 2015; originally announced February 2015.

Comments: 53 pages, 14 tables; v4 contains corrections to equations (1.2) and (3.1) and the proof of Lemma 3.2; note the published version may differ slightly

MSC Class: 11R42; 11M41

Journal ref: J. Number Theory 162 (2016) 312-375

A Minimax Bias Estimator for OLS Variances under Heteroskedasticity

Authors: Mumtaz Ahmed, Asad Zaman

Abstract: Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics… ▽ More Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate. △ Less

Submitted 4 May, 2014; originally announced May 2014.

The distribution of points on superelliptic curves over finite fields

Authors: GilYoung Cheong, Melanie Matchett Wood, Azeem Zaman

Abstract: We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the… ▽ More We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the limits taken differ slightly and the resulting distributions are interestingly different. △ Less

Submitted 1 October, 2012; originally announced October 2012.

Report number: AIM 2012-58 MSC Class: 11G20; 11T55; 11R58; 14H25; 11R45; 11R20; 11T06