On mean values of character sums (Q2508571)

In the course of his work on bounds for character sums, \textit{D. A. Burgess} [Proc. Lond. Math. Soc. (3) 13, 524--536 (1963; Zbl 0123.04404)] investigated the mean values \[ W_r(\chi, k):= \sum^k_{x=1}\,\Biggl| \sum^h_{m=1} \chi(x+ m)\Biggr|^{2r}, \] where \(\chi\) is a character modulo \(k\). Later he also examined [Acta Arith. 79, No. 4, 313--332 (1997; Zbl 0879.11045)] the sum \[ S_r(p):= \sum_\chi W_r(\chi, p^3), \] where the sum is over non-principal characters with \(\chi^{p^2}= \chi_0\), showing in particular that \[ S_3(p)\ll p^5 h^4+ p^6 h^2, \] when \(h\leq p\). The goal of the present paper is to prove the estimates \[ S_3(p)\ll p^5 h^3+ p^4 h^5\quad\text{and}\quad S_4(p)\ll p^5 h^4+ p^4 h^7, \] when \(h\leq p\), and \[ S_4(p)\ll p^2 h^8+ p^{10} h, \] when \(h\geq p\). Moreover, it is shown that \[ W_4(\chi, p^3)\ll p^{3/2} h^8+ p^2 h^y+ p^3 h^4 \] for any primitive character modulo \(p^3\), providing that \(h\leq p\).