Mean value of the character sums over interval \([1,\frac q8)\) (Q1025700)

Let \(k\geq2\) be a fixed integer, and \(q>8\) an odd integer. The author studies the character sum \[ S(\chi)=\sum\limits_{1\leq a<\frac{q}{8}}\chi(a). \] More precisely, he studies the \((2k)\)-th moment of \(S(\chi)\) with \(\chi\) varies over the set of primitive even characters to the modulus \(q\). He proves the following. \[ \sum\limits_{\chi\text{ even and primitive}}|S(\chi)|^{2k}=J(q)q^k\sum\limits_{(n,2q)=1}\frac{a_k(n)}{n^2}+O(q^{k+\varepsilon}), \] where \(J(q)\) is the number of primitive even characters to the modulus \(q\), and the terms \(a_k(n)\) are explicitly given, and are bounded by \(O(n^{\varepsilon})\).