The divisor function along arithmetic progressions and binary cubic polynomials

TL;DR

This paper establishes new equidistribution estimates for the divisor function in specific arithmetic progressions and applies these results to derive asymptotic formulas for the divisor function over almost all moduli of exponent 2/3 and along a particular binary cubic polynomial.

Contribution

It introduces a novel equidistribution estimate for the divisor function in progressions with moduli having two small factors, enabling new asymptotic formulas.

Findings

Asymptotic formula for divisor function over almost all moduli of exponent 2/3

Asymptotic formula for divisor function along the binary cubic polynomial XY^2+1

New equidistribution estimate for divisor function in special arithmetic progressions

Abstract

We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic progressions to almost all moduli of exponent $2/3$. Second, we show an asymptotic formula for the divisor function along the nonhomogeneous binary cubic polynomial $X Y^2+1$.