On the exponent of distribution of the ternary divisor function

TL;DR

This paper improves the known bounds on the distribution of the ternary divisor function in arithmetic progressions, showing it is well-distributed up to larger moduli than previously established.

Contribution

It provides new lower bounds for the exponent of distribution of the ternary divisor function, enhancing previous results by Heath-Brown and Friedlander--Iwaniec.

Findings

Exponent of distribution is at least 1/2+1/46 for prime moduli.

Exponent increases to 1/2+1/34 when averaging over a fixed residue class.

Improves understanding of the distribution of divisor functions in arithmetic progressions.

Abstract

We show that the exponent of distribution of the ternary divisor function $d_3$ in arithmetic progressions to prime moduli is at least 1/2+1/46, improving results of Heath-Brown and Friedlander--Iwaniec. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to 1/2 +1/34.