On the mean value of complete sums of Dirichlet characters of polynomials (Q1596160)

The author specifies an exact exponent of the convergence rate for the mean value of a complete sum of Dirichlet characters modulo a power of a prime. The following assertion is proved: if \(\alpha<k-1\), \(k\geq 2\), then the series \[ W = \sum\limits_k \sum\limits^{p^k}_{a_0=1} \sum\limits^{p^{k-\alpha}}_{a_1=1}\dots \sum\limits^{p^{k-\alpha}}_{a_n=1} \Biggl|\sum\limits^{p^{k-\alpha}}_{x=1} \chi(a_0+p^\alpha(a_1x+\dots +a_nx^n))p^{-k}\bigg|^{2m} \] converges for \(2m>\frac{n(n+1)}{2}+n+1\) and diverges for \(2m\leq\frac{n(n+1)}{2}+n+1\).