Character sums over shifted primes

TL;DR

This paper establishes a new bound for character sums over shifted primes involving the von Mangoldt function, extending previous results and improving the understanding of such sums in number theory.

Contribution

It provides a novel bound for sums of the form  ext{sum}_{n extless N}\u03a6(n)(n+a) with primitive characters, extending prior work by Friedlander, Gong, and Shparlinski.

Findings

New bound for character sums over shifted primes

Extended the range of previous results

Improved understanding of prime-related character sums

Abstract

For integer $q$, let $\chi$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}\Lambda(n)\chi(n+a),$$ where $\Lambda(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski