Triple correlations of ternary divisor functions. II

Abstract

Let [Formula: see text] denote the ternary divisor function. By introducing a simple argument, we prove an asymptotic formula for the triple correlation [Formula: see text] averaging over a short variable [Formula: see text], with an error term saving an arbitrary power of the logarithm over the trivial bound. We also prove an asymptotic formula by taking one of the ternary divisor functions to be the divisor function [Formula: see text], which substantially improves our previous results.