High order moments of character sums

Abstract

We establish the upper bound 1 p − 1 ∑ χ ≠ χ o | ∑ x = a + 1 a + B χ ( x ) | 2 k ≪ ϵ , k p k − 1 + ϵ + B k p ϵ , \begin{equation*}\frac {1}{p-1} \sum _{\chi \ne \chi _{o}}\big | \sum _{x=a+1}^{a+B} \chi (x) \big |^{2k} \ll _{\epsilon ,k} p^{k-1 +\epsilon } + B^{k} p^{\epsilon }, \end{equation*} with p p a prime and k k any positive integer, the sum being over all nonprincipal multiplicative characters ( mod p ) \pmod p .