A note on Burgess's estimate
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 \[ \left|\sum_{1\leq n\leq h}\left(\frac{n+N}{p}\right)\right|. \]
scientific article; zbMATH DE number 5980674
scientific article