ID: math/0503113

Large character sums: Pretentious characters and the Polya-Vinogradov Theorem

March 6, 2005

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Large sums of high order characters II

May 1, 2024

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Alexander P. Mangerel, Yichen You
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Let $\chi$ be a primitive character modulo $q$, and let $\delta > 0$. Assuming that $\chi$ has large order $d$, for any $d$th root of unity $\alpha$ we obtain non-trivial upper bounds for the number of $n \leq x$ such that $\chi(n) = \alpha$, provided $x > q^{\delta}$. This improves upon a previous result of the first author by removing restrictions on $q$ and $d$. As a corollary, we deduce that if the largest prime factor of $d$ satisfies $P^+(d) \to \infty$ then the level s...

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Large values of $L(1,\chi)$ for $k$-th order characters $\chi$ and applications to character sums

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Youness Lamzouri
Number Theory

For any given integer $k\geq 2$ we prove the existence of infinitely many $q$ and characters $ \chi\pmod q$ of order $k$, such that $|L(1,\chi)|\geq (e^{\gamma}+o(1))\log\log q$. We believe this bound to be best possible. When the order $k$ is even, we obtain similar results for $L(1,\chi)$ and $L(1,\chi\xi)$ where $\chi$ is restricted to even (or odd) characters of order $k$, and $\xi$ is a fixed quadratic character. As an application of these results, we exhibit large even ...

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Structure of large quadratic character sums

June 10, 2023

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Zikang Dong, Weijia Wang, Hao Zhang
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In this article, we study the distribution of large quadratic character sums. Based on the recent work of Lamzouri~\cite{La2022}, we obtain the structure results of quadratic characters with large character sums.

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The distribution of large quadratic character sums and applications

December 6, 2022

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Youness Lamzouri
Number Theory

In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to $x$, and deduce the interesting conseq...

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On Character Sums and Exponential Sums over Generalized Arithmetic Progressions

June 4, 2012

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Xuancheng Shao
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We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.

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Short Character Sums and the P\'{o}lya-Vinogradov Inequality

May 22, 2019

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Alexander P. Mangerel
Number Theory

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the P\'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q)$$ then for any $\epsilon > 0$ one may exhibit cancellation in partial sums of $\chi$ on the interval $[1,t]$ whenever $t > q^{\epsilon}$, i.e.,$$\sum_{n \leq t} \chi(n) = o_{q \righ...

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Three conjectures about character sums

December 23, 2021

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Andrew Granville, Alexander P. Mangerel
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We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estimate for short character sums, and upper bounds for $L(1,\chi)$ and $L(1+it,\chi)$) are more-or-less "equivalent". We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

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Moments of character sums to composite modulus

April 9, 2019

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Bryce Kerr
Number Theory

In this paper we consider the problem of estimating character sums to composite modulus and obtain some progress towards removing the cubefree restriction in the Burgess bound. Our approach is to estimate high order moments of character sums in terms of solutions to congruences with Kloosterman fractions and we deal with this problem by extending some techniques of Bourgain, Garaev, Konyagin and Shparlinski and Bourgain and Garaev from the setting of prime modulus to composit...

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Short character sums for composite moduli

December 31, 2011

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Mei-Chu Chang
Number Theory

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q'=\prod_{p|q}p$ of the modulus $q$ satisfies $\log q'\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\rm{P\acute{o}lya}$ and Vinogradov inequalities are indicated.

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Burgess bounds for short mixed character sums

April 7, 2014

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D. R. Heath-Brown, L. B. Pierce
Number Theory

This paper proves nontrivial bounds for short mixed character sums by introducing estimates for Vinogradov's mean value theorem into a version of the Burgess method.

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