On Karatsuba's Problem Concerning the Divisor Function $\tau(n)$

TL;DR

This paper investigates the asymptotic behavior of a sum involving the divisor function, specifically analyzing the ratio of divisor counts for consecutive integers offset by a fixed integer.

Contribution

It provides new asymptotic results for sums involving the divisor function ratios, addressing a problem related to Karatsuba's conjecture.

Findings

Derived asymptotic formulas for the sum involving divisor function ratios

Extended understanding of divisor function behavior in shifted sums

Provided bounds and estimates for the sum as x tends to infinity

Abstract

We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}$. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.