A triple convolution sum of the divisor function

TL;DR

This paper investigates the asymptotic behavior of a triple convolution sum involving the divisor function, providing explicit lower bounds and a new framework to estimate the conjectured constant, advancing understanding of divisor sum asymptotics.

Contribution

It introduces a novel theoretical framework using Tauberian theory for multiple Dirichlet series to estimate the sum and predict the constant in Browning's conjecture.

Findings

Established an explicit lower bound for the sum.

Provided a new approach to estimate the conjectured constant.

Connected algebraic and geometric considerations with analytic methods.

Abstract

We study the triple convolution sum of the divisor function given by $$\sum_{n\leq x} d(n)d(n-h)d(n+h)$$ for $h\neq 0$ and $d(n)$ denotes the number of positive divisors of $n$. Based on algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to $c_hx(\log x)^3$, for a suitable constant $c_h\neq 0$, as $x\to \infty$. This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this paper, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant $c_h$.