Character sums with subsequence sums

Abstract

Let χ be a primitive multiplicative character modulo an integer m ≥ 1. Using some classical bounds of character sums, we estimate the average value of the character sums with subsequence sums \( T_m (\mathcal{S},\chi ) = \sum\nolimits_{\mathcal{I} \subseteq \{ 1, \ldots ,N\} } {\chi (\sum\nolimits_{i \in \mathcal{I}} {s_i } )} \) taken over all N-element sequences S = (s1, …, sN) of integer elements in a given interval [K + 1, K + L]. In particular, we show that Tm(S, χ) is small on average over all such sequences. We apply it to estimating the number of perfect squares in subsequence sums in almost all sequences.