arXiv subjects

William Banks

Publications and source records attributed to William Banks.

10 recordsLinked to original sources

arXiv2026

Unshared zeros of Dirichlet $L$-functions

We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear combination of Dirichlet $L$-functions) can vanish at every zero of a fixed $L(s,\chi_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. Since distinct primitive characters are distinguished by their values, varying $x$ isolates the contribution of each $L$-function within a linear combination, and we deduce that for a positive density of $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\chi_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vall\'ee Poussin zero-free region.

math.NT↗

arXiv2026

Sets of integers satisfying Bateman-Horn statistics

In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n \le x for which f_1(n), ..., f_k(n) are all prime, where (f_1, ..., f_k) is an admissible k-tuple of polynomials in one variable. We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced "f_1(n), ..., f_k(n) are all prime" with "f_1(n), ..., f_k(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful.

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arXiv2023

Descriptive properties of the type of an irrational number

The type $τ$($α$) of an irrational number $α$ measures the extent to which rational numbers can closely approximate $α$. More precisely, $τ$($α$) is the infimum over those t$\in$R for which |$α$--h/k| 0. In this paper, we regard the type as a function $τ$:R\Q$\rightarrow$[1,$\infty$] and explore its descriptive properties. We show that $τ$ is invariant under the natural action of GL2(Q) on R\Q. We show that $τ$ is densely onto, and we compute the descriptive complexity of the pre-image of the singletons and of certain intervals. Finally, we show that the function $τ$ is [1,$\infty$]-upper semi-Baire class 1 complete.

math.GN↗

arXiv2022

On the greatest common divisor of integer parts of polynomials

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error terms in our formulas are of various strengths depending on the Diophantine properties of the leading coefficients of these polynomials.

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arXiv2020

On a conjecture of Soundararajan

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,χ)$ for characters $χ$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,χ)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.

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arXiv2019

Symmetric primes revisited

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a symmetric pair. In the present paper, that theorem is improved to what is likely to be the best possible result. We also establish that there exist infinitely many symmetric pairs of primes. In fact, we show that for every integer m at least 2 there is a string of m consecutive primes, any two of which form a symmetric pair.

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arXiv2019

Large prime gaps and probabilistic models

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

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arXiv2019

Congruences with intervals and arbitrary sets

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski, Michel and Sawin (2018).

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arXiv2016

Bounds on short character sums and L-functions for characters with a smooth modulus

We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus $q$ has a small core $\prod_{p\mid q}p$. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding $L$-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the region free of Siegel zeros.

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arXiv2012

Self-intersections of the Riemann zeta function on the critical line

We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z in C the equation ζ(1/2+it)=z has at most one solution t in R. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R, then either the set {(a,b) in R^2:a \ne b and F(a)=F(b)} is countable, or the image F(R) is a loop in C.

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