On high power moments of the error term of the Dirichlet divisor function over primes

Abstract

Let $3⩽k⩽9$ be a fixed integer, $p$ be a prime and $d(n)$ denote the Dirichlet divisor function. We use $\Delta (x)$ to denote the error term in the asymptotic formula of the summatory function of $d(n)$. The aim of this paper is to study the $k$-th power moments of $\Delta (p)$, namely ${\sum }_{p⩽x}{\Delta }^k(p)$, and we give an asymptotic formula.

15 pages. arXiv admin note: substantial text overlap with arXiv:2410.00329