Abstract
In this series we examine the calculation of the 2kth 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 T2 with divisor functions as coefficients.
Highlights
This paper is part 3 of a sequence of papers devoted to understanding how to conjecture all of the integral moments of the Riemann zeta-function from a number theoretic perspective
There will be many off-diagonal terms and it is the care of these that is the concern of these papers
In particular it is necessary to treat the off-diagonal terms by a method invented by Bogomolny and Keating [1,2]
Summary
This paper is part 3 of a sequence of papers devoted to understanding how to conjecture all of the integral moments of the Riemann zeta-function from a number theoretic perspective. Keating / Indagationes Mathematicae 26 (2015) 736–747 convolution problem but we try to solve it precisely in a way that exhibits all of the main terms of the expected formula Our treatment is not rigorous; in particular we conjecture the shape of the fundamental shifted convolution at a critical juncture This is to be expected since for example no one knows how to evaluate τ3(n)τ3(n + 1). The formula we obtain is in complete agreement with all of the main terms predicted by the recipe of [3] (and in particular, with the leading order term conjectured in [9])
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