A note on Burgess's estimate
type: Item
instance of: scholarly article
title: A note on Burgess's estimate
zbMATH Open document ID: 1271.11085
DOI: 10.1134/S000143461009004X
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: This is an expository account of \textit{D. A. Burgess}'s bound [Mathematika 4, 106--112 (1957; Zbl 0081.27101)]. For simplicity the authors consider only the most basic case, proving that if \(p\) is a prime then \[ \sum_{1\leq n\leq H}\left(\frac{n}{p}\right)\ll H^{1-1/r}p^{(r+1)/(4r^2)}\log p \] uniformly for any integer \(r\geq 1\). The main theme in the exposition is that Burgess's original proof involved a number of distinct techniques interlaced within the argument. In the present paper the authors separate these out, so as to make the reasoning considerably clearer. In particular they show how the ``Rademacher--Menchov device'' [\textit{H. Rademacher}, Math. Ann. 87, 112--138 (1922; JFM 48.0485.05)] and [\textit{D. Menchov}, Fund. Math. 4, 82--105 (1923; JFM 49.0293.01)] allows one to pass efficiently from moments of \[ \max_{h\leq H}\left|\sum_{1\leq n\leq h}\left(\frac{n+N}{p}\right)\right| \] to the corresponding moments of \[ \left|\sum_{1\leq n\leq h}\left(\frac{n+N}{p}\right)\right|. \]
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 5980674
zbMATH Keywords: Burgess's estimate
zbMATH Keywords: character sum
zbMATH Keywords: Rademacher--Menchov device
zbMATH Keywords: real character
author: Patrick Gallagher
author: Hugh L Montgomery
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5980674