Averages of real character sums (Q1125403)

Real non-principal characters are representable as \(\chi _D (n) =\left (\frac Dn \right)\) in terms of the Kronecker symbol with \(D\) running over non-square quadratic discriminants. Let \(S _D (Y)=\sum _{n\leq Y} \chi _D (n)\). The main results in this papers are: \begin{align*} &\sum _{|D|\leq X} \max _Y |S_D (Y) |^{2k} \ll _k X ^{k+1},\tag{i}\\ &\sum _{|D|\leq X}|S_D (Y)|^2 \ll XY \log X. \tag{ii} \end{align*} The estimate (i) is a generalization of a theorem of \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Can. J. Math. 31, 476-487 (1979; Zbl 0416.10030)] concerning sums of Legendre symbols. The estimate (ii) is an improvement, as to the exponent of the factor \(\log X\), of a theorem of the reviewer [J. Number Theory 5, 203-214 (1973; Zbl 0257.10015)]. Obtaining the sharp logarithmic factor is indeed the main delicacy here, and for this purpose the author appeals to an approximate formula due to \textit{J. D. Vaaler} [Bull. Am. Math. Soc. 12, 183-216 (1985; Zbl 0575.42003)] for the characteristic function of the interval \([0,Y]\).