Bounds on short character sums and L-functions with characters to a powerful modulus
Bounds on short character sums and L-functions with characters to a powerful modulus
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短字符和和具有强大模数的字符的 L 函数的界限
DOI：
10.1007/s11854-019-0060-4
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
I. Shparlinski
中科院分区：
文献类型：
--
作者：
W. Banks;I. Shparlinski
We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus q has a small core Πp|qp. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding L-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the classical zero-free region (which might include Siegel zeros). Finally, we improve the previous best value L=125=2.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=\frac{12}{5}=2.2$$\end{document} of the Linnik constant for primes in arithmetic progressions modulo powers of a fixed prime to L < 2.1115.
We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus q has a small core Πp|qp. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding L-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the classical zero-free region (which might include Siegel zeros). Finally, we improve the previous best value L=125=2.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=\frac{12}{5}=2.2$$\end{document} of the Linnik constant for primes in arithmetic progressions modulo powers of a fixed prime to L < 2.1115.