On a sum of the error term of the Dirichlet divisor function over primes

TL;DR

This paper derives an asymptotic formula for the sum of the squared error term of the Dirichlet divisor function over primes, advancing understanding of error distribution in divisor sums at prime arguments.

Contribution

It introduces an asymptotic formula for the sum of squared error terms of the divisor function evaluated at prime numbers, a novel result in analytic number theory.

Findings

Established an asymptotic formula for the sum over primes of the squared error term.

Enhanced understanding of the distribution of the divisor function's error term at prime points.

Abstract

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^2(p)$, where $p$ runs over primes. We prove that there exists an asymptotic formula.