The maximum size of short character sums

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Abstract

In the present note, we prove new lower bounds on large values of character sums $\Delta(x,q):=\max_{\chi \neq \chi_0} \vert \sum_{n\leq x} \chi(n)\vert$ in certain ranges of $x$. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of $L$- functions, we improve some results of Hough in the range $\log x = o(\sqrt{\log q})$. Our results are expressed using the counting function of $y$- friable integers less than $x$ where we improve the level of smoothness $y$ for short intervals.
Original languageUndefined/Unknown
JournalThe Ramanujan journal
Publication statusAccepted/In press - 18 May 2018

Keywords

  • math.NT
  • 11L40, 11N25

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