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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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Volume 27, Issue 5

# Character sums over unions of intervals

Xuancheng Shao
Published Online: 2014-01-10 | DOI: https://doi.org/10.1515/forum-2013-0080

## Abstract

Let q be a cube-free positive integer and $\chi \phantom{\rule{3.33333pt}{0ex}}\left(\text{mod}\phantom{\rule{3.33333pt}{0ex}}q\right)$ be a non-principal Dirichlet character. Our main result is a Burgess-type estimate for ${\sum }_{n\in A}\chi \left(n\right)$, where $A\subset \left[1,q\right]$ is the union of s disjoint intervals ${I}_{1},...,{I}_{s}$. We obtain a nontrivial estimate for the character sum over A whenever $|A|{s}^{-1/2}>{q}^{1/4+ϵ}$ and each interval Ij ($1\le j\le s$) has length $|{I}_{j}|>{q}^{ϵ}$ for any $ϵ>0$. This follows from an improvement of a mean value Burgess-type estimate studied by Heath-Brown [Number Theory and Related Fields, Springer Proc. Math. Statist. 43, New York (2013), 199–213].

Keywords: Character sum; harmonic analysis

MSC: 11L40

Revised: 2013-10-24

Published Online: 2014-01-10

Published in Print: 2015-09-01

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 3017–3026, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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MEI-CHU CHANG and IGOR E. SHPARLINSKI
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