A generalization of the Titchmarsh divisor problem

TL;DR

This paper extends the Titchmarsh divisor problem by deriving an asymptotic formula for sums involving the $k$-free divisor function over primes shifted by a fixed integer, using advanced analytic techniques.

Contribution

It introduces a generalized asymptotic formula for a class of arithmetic functions, including the $k$-free divisor function, related to prime sums shifted by an integer.

Findings

Established an asymptotic formula for $ extstyle \sum_{p ext{ prime}} ext{} d^{(k)}(p-a)$

Applied a result of Felix to derive the formula

Generalized the approach to include various arithmetic functions

Abstract

Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \begin{equation*} \sum_{p\leq x} d^{(k)}(p-a) =b_k \cdot x+O\left(\frac{x}{\log x}\right) \end{equation*} related to the Titchmarsh divisor problem, where $b_k$ is a positive constant dependent on $k$ and $a$. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, $k$-free divisor function and the proper Pillai's function.