Bounds and Conjectures for additive divisor sums

TL;DR

This paper investigates bounds and conjectures for additive divisor sums, which are crucial in understanding moments of the Riemann zeta function, providing new lower bounds, conjectural formulas, and connections to probabilistic methods.

Contribution

It establishes a lower bound of correct order for additive divisor sums and explores conjectural asymptotics using novel probabilistic and analytical approaches.

Findings

Established a lower bound of correct order for $D_{k,\, ext{ell}}(x,h)$

Simplified the leading term in the conjectural asymptotic formula

Connected probabilistic methods with recent results by Terry Tao

Abstract

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} \tau_k(n) \tau_{\ell}(n+h)$ where $\tau_k$ and $\tau_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main result is a lower bound of the correct order of magnitude for $D_{k,\ell}(x,h)$, uniform in $h$. In addition, the conjectural asymptotic formula for $D_{k,\ell}(x,h)$ is studied. Using an argument of Ivi\'{c} and Conrey-Gonek the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao.