The maximum size of short character sums
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The maximum size of short character sums Marc Munsch1 Received: 18 May 2018 / Accepted: 31 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In the present note, we prove new lower bounds on large values of character sums Δ(x, q) := maxχ =χ0 n≤x χ (n) 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 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. Keywords Dirichlet characters · Large values · Friable numbers · Multiplicative functions Mathematics Subject Classification 11L40 · 11N25
1 Introduction The behavior of character sums Sχ (x) := n≤x χ (n), where χ is a non-principal Dirichlet character modulo q, is of great importance in many number theoretical problems such as the distribution of non-quadratic residues or primitive roots. Showing some cancellation in such character sums has been an intensive topic of study for many decades, originating from the unconditional bound of Pólya and Vinogradov √ S(χ ) q log q. In the present note, we are interested in the opposite problem which consists of bounding from below the quantity (1) Δ(x, q) := max χ (n) . χ =χ0 n≤x
The author is supported by the Austrian Science Fund (FWF) Project Y-901 “Probabilistic methods in analysis and number theory” led by Christoph Aistleitner.
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Marc Munsch [email protected] 5010 Institut für Analysis und Zahlentheorie, Steyrergasse 30, 8010 Graz, Austria
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M. Munsch
Here and throughout this paper we write log j for the j-th iterated logarithm, for example log2 q = log log q. Assuming the Generalized Riemann Hypothesis, Mont√ gomery and Vaughan showed Δ(x, q) q log2 q (strengthening the work of Pólya and Vinogradov) which matches the omega results obtained earlier by Paley [15]. Nonetheless, the situation for shorter intervals remains in certain cases open. An intensive study of this quantity through the computation of high moments was carried out by Granville and Soundararajan [8] giving very precise results for short intervals. A few years later, Soundararajan developed the so-called resonance method [16] in order to show the √ existence of large values of L-functions at the central point. In the intermediate range log q < log x < (1 − ) log q, even though the situation remains widely open, recent progress have been made using this method (see the results obtained by Hough [11] reinforcing the previous results of Granville and Soundararajan [8]). It is worth noticing that de la Bretèche and Tenenbaum [4] recently obtained the following result improving earlier bounds of Hough for very large x, precisely as soon as log x ≥ (log q)1/2+ , Δ(x, q) ≥
√
log(q/x) log3 (q/x) x exp ( 2 + o(1)) . log2 (q/x) √
√ In this paper, we will give new bounds in the rang
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