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High order moments of character sums

Author(s): Todd Cochrane; Zhiyong Zheng
Journal: Proc. Amer. Math. Soc. 126 (1998), 951-956.
MSC (1991): Primary 11L40, 11D79
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Abstract: We establish the upper bound

$\begin{equation*}\frac{1}{p-1} \sum _{\chi \ne \chi _{o}}\big | \sum _{x=a+1}^{a+B} \chi (x) \big |^{2k} \ll _{\epsilon ,k} p^{k-1 +\epsilon } + B^{k} p^{\epsilon }, \end{equation*}$

with $p$ a prime and $k$ any positive integer, the sum being over all nonprincipal multiplicative characters $\pmod p$ .

References:

[1]: A. Ayyad, T. Cochrane and Z. Zheng, The congruence $x_{1}x_{2} \equiv x_{3}x_{4} \pmod p$ , the equation $x_{1}x_{2}=x_{3}x_{4}$ and mean values of character sums, J. Number Theory 59 (2) (1996), 398-413. MR 97i:11091
[2]: D.A. Burgess, On character sums and L-series. II, Proc. London Math. Soc. (3) 13 (1963), 524-536. MR 26:6133
[3]: H.L. Montgomery and R.C. Vaughan, Exponential sums with multiplicative coefficients, Inventiones Math. 43 (1977), 69-82. MR 56:15579
[4]: H.L. Montgomery and R.C. Vaughan, Mean values of character sums, Canad. J. Math. 31 (3) (1979), 476-587. MR 81c:10043


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