2018•Proceedings of the London Mathematical SocietyOpen access

Moments of zeta and correlations of divisor‐sums: V

J. Brian Conrey, Jonathan P. Keating

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Abstract

In this series of papers we examine the calculation of the 2 k 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 , 2 T ] of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.

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In this series of papers we examine the calculation of the 2 k 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 , 2 T ] of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.

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Available abstract

In this series of papers we examine the calculation of the 2 k 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 , 2 T ] of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.

Key concepts: Divisor (algebraic geometry), Mathematics, Statistics, Combinatorics

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