Alexander Walker

activity

20132025

most citedSecond Moments in the Generalized Gauss Circle Problem

21 citations · 52 across the 14 of their papers we have counts for

collaborators

17 papers

math.NT2025

The Fibonacci Zeta Function and Modular Forms

Eran Assaf, Chan Ieong Kuan, David Lowry-Duda +1

We show that a family of Dirichlet series generalizing the Fibonacci zeta function $\sum F (n)^{- s}$ has meromorphic continuation in terms of dihedral $GL (2)$ Maass forms.

math.NT2024

The Fibonacci Zeta Function and Continuation

Eran Assaf, Chan Ieong Kuan, David Lowry-Duda +1

We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F (n)^{- s}$ , where $F (n)$ denotes the…

math.NT2024★ 1 cited

Self-Correlations of Hurwitz Class Numbers

Alexander Walker

The asymptotic study of class numbers of binary quadratic forms is a foundational problem in arithmetic statistics. Here, we investigate finer statistics of class numbers by studyi…

math.NT2023

Counting Divisors in the Outputs of a Binary Quadratic Form

Chan Ieong Kuan, David Lowry-Duda, Alexander Walker +1

For a fixed natural number $h$ , we prove meromorphic continuation of the two-variable Dirichlet series $\sum_{m} r_{2} (m) σ_{w} (m + h) (m + h)^{- s + w}$ to $C^{2}$ and use this to…

math.NT2023

Sums of Cusp Form Coefficients Along Quadratic Sequences

Chan Ieong Kuan, David Lowry-Duda, Alexander Walker +1

Let $f (z) = \sum A (n) n^{(k - 1) /2} e (n z)$ be a cusp form of weight $k \geq 3$ on $Γ_{0} (N)$ with character $χ$ . By studying a certain shifted convolution sum, we prove that $\sum_{n \…

math.NT2022

Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions

Alexander Walker

We give conditions under which certain digit-restricted integer sets avoid $k$ -term arithmetic progressions. These sets and their harmonic sums can be computed efficiently. Through…

math.NT2020

Arithmetic Progressions of Squares and Multiple Dirichlet Series

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic conti…

math.NT2019★ 6 cited

Triple Correlation Sums of Coefficients of Cusp Forms

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a (h) b (m) c (2 m - h)$ , in which $a (n), b (n), c (n)$ are un-normalized Fourier coefficients of holomorphic cusp…

math.NT2018

Arithmetic Progressions with Restricted Digits

Aled Walker, Alexander Walker

For an integer $b ⩾ 2$ and a set $S \subset {0, \dots, b - 1}$ , we define the Kempner set $K (S, b)$ to be the set of all non-negative integers whose base- $b$ digital…

math.NT2018★ 4 cited

A Shifted Sum for the Congruent Number Problem

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$ . The asymptotic growth of this series depends explicitly on whether or not $t$ is a…

math.NT2017★ 4 cited

The Laplace Transform of the Second Moment in the Gauss Circle Problem

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

The Gauss circle problem concerns the difference $P_{2} (n)$ between the area of a circle of radius $n$ and the number of lattice points it contains. In this paper, we study th…

math.NT2017★ 21 cited

Second Moments in the Generalized Gauss Circle Problem

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k} (n)^{2}$ , where $P_{k} (n)$ is the discrepa…

math.NT2016★ 6 cited

Sign Changes of Coefficients and Sums of Coefficients of L-Functions

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is c…

math.GT2015

Graphs of Large Girth and Surfaces of Large Systole

Bram Petri, Alexander Walker

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinit…

math.NT2015★ 3 cited

Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_{f} (n)$ denote the sum of the first $n$ Fourier coefficients of $f$ . In analogy with Dirichlet's divisor proble…

math.NT2015★ 7 cited

The Second Moment of Sums of Coefficients of Cusp Forms

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda +1

Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_{f} (n)$ and $S_{g} (n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivic [HI], building on Chandr…

math.NT2013

Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences

Zane Kun Li, Alexander W. Walker

The coefficient series of the holomorphic Picard-Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These…