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

TL;DR

This paper investigates the high power moments of the error term in the Dirichlet divisor function evaluated at primes, providing asymptotic formulas for these moments.

Contribution

It derives new asymptotic formulas for the k-th power moments of the error term of the divisor function at primes for 3 ≤ k ≤ 9.

Findings

Asymptotic formulas for moments of the error term at primes

Results valid for fixed integers 3 ≤ k ≤ 9

Enhanced understanding of divisor function error behavior over primes

Abstract

Let $3\leqslant k\leqslant9$ 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 $\sum_{p\leqslant x}\Delta^k(p)$, and we give an asymptotic formula.