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On the mean value of complete sums of Dirichlet characters of polynomials. (English. Russian original) Zbl 0967.11031

Mosc. Univ. Math. Bull. 54, No. 4, 45-46 (1999); translation from Vestn. Mosk. Univ., Ser I 1999, No. 4, 64-65 (1999).
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 \(2m\leq\frac{n(n+1)}{2}+n+1\).

MSC:

11L40 Estimates on character sums