Circle and divisor problems, and double series of Bessel functions
type: Item
instance of: scholarly article
title: Circle and divisor problems, and double series of Bessel functions
zbMATH Open document ID: 1326.11059
author: Yanyan Li
published in: Advances in Mathematics
publication date: 2013-04-05T00:00:00Z
review text: In this very intricate paper (a typical example of ``hard analysis''), the authors -- for the first time in history -- give a proof of an identity due to \textit{S. Ramanujan} [The Lost Notebook and other unpublished papers. With an introduction by George E. Andrews. New Delhi: Narosa Publishing House; Berlin: Springer-Verlag (1988; Zbl 0639.01023), p. 335]: Entry 1.1. If \(0<\theta<1,\,x>0\), and \(F(x)\) as defined below, then \[ \begin{multlined} \sum_{n=1}^{\infty}\,F\left({x\over n}\right)\sin{(2\pi n\theta)}=\pi x\left({1\over 2-\theta}\right)-{1\over 4}\cot{(\pi\theta)}\cr +{1\over 2}\sqrt{x}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}\,\left\{{J_1(4\pi\sqrt{m(n+\theta)x}\over\sqrt{m(n+\theta)}} - {J_1(4\pi\sqrt{m(n+1-\theta)x}\over\sqrt{m(n+1-\theta)}}\right\}.\end{multlined} \] Here \(J_1\) denotes the ordinary Bessel function of order \(1\) and \[ F(x)=\begin{cases} [x], & \text{if }x\text{ is not an integer},\\ x-{1\over 2}, &\text{if }x\text{ is an integer},\end{cases} \] where \([x]\) is the integer part of \(x\). The main tool is an important theorem of the analytic continuation of a family of Dirichlet series \[ G(x,\theta,s)=\sum_{m=1}^{\infty}\,{a(x,\theta,m)\over m^s}, \] with coefficients \[ a(x,\theta,m)=\sum_{n=0}^{\infty}\,\left\{{J_1(4\pi\sqrt{m(n+\theta)x}\over\sqrt{m(n+\theta)}} - {J_1(4\pi\sqrt{m(n+1-\theta)x}\over\sqrt{m(n+1-\theta)}}\right\}, \] with fixed \(x>0\) and for any \(0<\theta<1\). Crucial is the following theorem. Theorem 1.2. For any \(x>0\) and any \(0<\theta<1\), \(G(x,\theta,s)\) has an analytic continuation to the half plane \(\text{Re}\,(s)>{8\over 17}\). For \(s\) in this half plane, and \(x>0\), the series for \(G\) converges uniformly with respect to \(\theta\) in any compact subinterval of \((0,1))\). If \(x\) is not an integer, the above holds in the larger half plane \(\text{Re}\,(s)>{1\over 3}\). This theorem, and some very clever calculation methods (which probably can be very useful in other situations connected with t…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 6150708
zbMATH Keywords: circle problem
zbMATH Keywords: Bessel functions
zbMATH Keywords: Ramanujan's lost notebook
MaRDI profile type: Publication
full work available at URL: j.aim.2012.11.017
OpenAlex ID: W2001949621
DOI: 10.1016/J.AIM.2012.11.017
description: scientific article; zbMATH DE number 6150708