On Karatsuba’s problem concerning the divisor function

Abstract

We study an asymptotic behavior of the sum n x τ (n) τ (n + a). Here τ (n) denotes the number of divisors of n and a 1 is a fixed integer.

On the Karatsuba divisor problem

“…Note that the solution of Titchmarsh divisor problem is based on Bombieri-Vinogradov theorem. In our paper we essentially use the analogue of Bombieri-Vinogradov theorem, obtained by Korolev (see [9], Lemma 13). Note that the methods of this paper can also be applied for other sums.…”

“…Remark. Using Lemma 6 and some estimates following from inequality (4), one can improve the dependence on d in the remainder term in (9).…”

On an asymptotic behavior of the divisor function $τ(n)$

“…[1] 4. Worth mentioning the result in [5] concerning Karatsuba's problem on determining the asymptotic behavior of the sum…”