On the Restricted Divisor Function in Arithmetic Progressions

TL;DR

This paper provides new asymptotic estimates for sums of a restricted divisor function over short arithmetic progressions, advancing understanding related to the pair correlation of fractional parts of quadratic functions.

Contribution

It introduces improved asymptotic estimates for the restricted divisor function sums in arithmetic progressions, connecting to pair correlation problems.

Findings

Enhanced asymptotic bounds for divisor sums

Connections to quadratic fractional part pair correlation

Improved results over previous work by J. Truelsen

Abstract

We obtain several asymptotic estimates for the sums of the restricted divisor function $$ \tau_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function $\alpha k^2$, $k=1,2,...$ with a real $\alpha$.