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  • 1 Macquarie University Department of Computing Sydney NSW 2109 Australia
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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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$T_m (\mathcal{S},\chi ) = \sum\nolimits_{\mathcal{I} \subseteq \{ 1, \ldots ,N\} } {\chi (\sum\nolimits_{i \in \mathcal{I}} {s_i } )}$$ \end{document}
taken over all N-element sequences S = (s 1, …, s N) of integer elements in a given interval [K + 1, K + L]. In particular, we show that T m(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.