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Brian Conrey

Publications and source records attributed to Brian Conrey.

27 records · Page 2Linked to original sources
arXiv2015

Moments of zeta and correlations of divisor-sums: III

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 is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as 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 $T^2$ with divisor functions as coefficients.

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arXiv2015

Moments of zeta and correlations of divisor-sums: I

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

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arXiv2013

Period functions and cotangent sums

We investigate the period function of $\sum_{n=1}^\inftyσ_a(n)\e{nz}$, showing it can be analytically continued to $|\arg z|<π$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In particular, we find a reciprocity formula for the Vasyunin sum.

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arXiv2012

Smooth neighbors

We give a new algorithm that quickly finds smooth neighbors.

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arXiv2011

Asymptotic Large Sieve

Motivated by applications to the study of L-functions, we develop an asymptotic version of the large sieve inequality for linear forms in primitive Dirichlet characters.

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arXiv2011

Critical zeros of Dirichlet $L$-functions

We use the Asymptotic Large Sieve and Levinson's method to obtain lower bounds for the proportion of simple zeros on the critical line of the twists by primitive Dirichlet characters of a fixed L-function of degree 1,2, or 3.

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arXiv2007

Autocorrelation of ratios of L-functions

We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of L-functions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.

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arXiv2005

Pseudomoments of the Riemann zeta-function and pseudomagic squares

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix.

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