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

Abstract

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

32 pages