## Character sums over shifted primes

Bryce Kerr
Maquary University

Abstract

For integer $q$, let $\chi$ be a primitive character of the multiplicative group $\left(\mathbb{Z}/q\mathbb{Z}\right)^{*}$. For any integer $a$ coprime to $q$, we consider bounding the sums $$\sum_{p\le N}\chi(p+a)$$ where the above sum is over all prime numbers less than $N$. We show cancellation in such sum occurs in the range $N\ge q^{5/6+\varepsilon}$ which improves a result of Friedlander, Gong and Shparlinski.