Searcharxiv⌕ Search

arXiv subjects

Brian Conrey

Publications and source records attributed to Brian Conrey.

At least 19 recordsLinked to original sources
arXiv2026

Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions

Let $\Pi_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, \Pi_{0}\,\times\, \chi)$ lie on the critical line, where $\chi$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $\Pi_{0}$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $L(s, \Pi_{0}\times \chi)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^{\epsilon}\le T \le Q^{1/3-\epsilon}$. When $T=Q^{\epsilon}$, our asymptotic formula allows Dirichlet polynomials of length $\theta <1/2-\epsilon$; when $\theta=0$, it gives a strong error term of size $O_{\epsilon}(Q^{7/4+\epsilon})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $L$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $\Pi_{0}$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.

math.NT↗
arXiv2025

Twisted moments of characteristic polynomials of random matrices in the unitary group

Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $\zeta(s)$. Their approach indicates how lower twisted moments of $\zeta(s)$ may be used to evaluate higher moments. In this paper, we present a rigorous random matrix theory analogue of their heuristic. To do this, we develop a notion of "twisted moment" of characteristic polynomials of matrices in the unitary group $U(N)$, and we prove several identities involving Schur polynomials. Our results may be viewed as a proof of concept of the heuristic for $\zeta(s)$.

math.NT↗
arXiv2024

Moments of the derivative of the characteristic polynomial of unitary matrices

Let $\Lambda_X(s)=\det(I-sX^{\dagger})$ be the characteristic polynomial of a Haar distributed unitary matrix $X$. It is believed that the distribution of values of $\Lambda_X(s)$ model the distribution of values of the Riemann zeta-function $\zeta(s)$. This principle motivates many avenues of study. Of particular interest is the behavior of $\Lambda_X'(s)$ and the distribution of its zeros (all of which lie inside or on the unit circle). In this article we present several identities for the moments of $\Lambda_X'(s)$ averaged over $U(N)$, for $s \in \mathbb{C}$ as well as specialized to $|s|=1$. Additionally, we prove, for positive integer $k$, that the polynomial $\int_{U(N)} |\Lambda_X(1)|^{2k} \mathrm{dX}$ of degree $k^2$ in $N$ divides the polynomial $\int_{U(N)} |\Lambda_X'(1)|^{2k} \mathrm{dX}$ which is of degree $k^2+2k$ in $N$ and that the ratio, $f(N,k)$, of these moments factors into linear factors modulo $4k-1$ if $4k-1$ is prime. We also discuss the relationship of these moments to a solution of a second order non-linear Painl\'{e}ve differential equation. Finally we give some formulas in terms of the $_3F_2$ hypergeometric series for the moments in the simplest case when $N=2$, and also study the radial distribution of the zeros of $\Lambda_X'(s)$ in that case.

math-ph↗
arXiv2023

Factoring determinants and applications to number theory

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin - Okounkov, Case - Geronimo, and Basor - Erhardt to prove that, in certain cases, these unitary averages factor as polynomials in the matrix size into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the ``swap'' terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from Random Matrix Theory.

math.NT↗
arXiv2022

Averages of quadratic twists of long Dirichlet polynomials

We investigate averages of long Dirichlet polynomials twisted by Kronecker symbols and we compare our result with the recipe of [CFKRS]. We are able to compute these averages in the case that the length of the polynomial is a power less than 2 of the basic scaling parameter on the assumption of the Lindel\"of Hypothesis for $L$-functions of quadratic characters, and we show that the answer is consistent with this recipe. This corresponds, in terms of the recipe, to verifying 0- and 1-swap terms.

math.NT↗
arXiv2022

Averages of long Dirichlet polynomials with modular coefficients

We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a \textit{long} Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

math.NT↗
arXiv2022

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

math.NT↗
arXiv2021

Which numbers are not the sum plus the product of three positive integers?

We investigate the number $R_3(n)$ of representations of $n$ as the sum plus the product of three positive integers. On average, $R_3(n)$ is $\frac{1}{2}\log^2 n$. We give an upper bound for $R_3(n)$ and an upper bound for the number of $n \leq N$ such that $R_3(n) = 0$. We conjecture that $R_3(n)= 0$ infinitely often.

math.NT↗
arXiv2018

Moments of zeta and correlations of divisor-sums: V

In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.

math.NT↗
arXiv2017

A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

math.NT↗
arXiv2017

Knots and ones

We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the Möbius function. We also prove a conjecture about further divisibility properties of the invariants.

math.GT↗
arXiv2016

Averages of ratios of the Riemann zeta-function and correlations of divisor sums

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

math.NT↗
arXiv2016

Moments of zeta and correlations of divisor-sums: IV

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^3$ with divisor functions as coefficients.

math.NT↗
arXiv2016

Intransitive Dice

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice $A_1,\dots,A_k$ are randomly selected. We conjecture that the probability of ties goes to 0 as $n$ grows. We conjecture and provide some supporting evidence that---contrary to intuition---each of the $2^{k \choose 2}$ assignments of $\succ$ or $\prec$ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select $k$ dice $A_1,\dots,A_k$ and observe that $A_1 \succ A_2 \succ \ldots \succ A_k$. Then our conjecture asserts that the outcomes $A_k \succ A_1$ and $A_1 \prec A_k$ both have probability approaching $1/2$ as $n \rightarrow \infty$.

math.CO↗