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authors – for the first time in history – give a proof of an identity due to S. Ramanujan [The Lost Notebook and other unpublished papers.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 \[ \begin{split} \sum_{n=1}^{\infty}\,F\left({x\over n}\right)\sin{(2\pi n\theta)}=\pi x\left({1\over 2-\theta}\right)-{1\over 4}\cot{(\pi\theta)}\cr +{1\over 2}\sqrt{x}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}\,\left\{{J_1(4\pi\sqrt{m(n+\theta)x}\over\sqrt{m(n+\theta)}} - {J_1(4\pi\sqrt{m(n+1-\theta)x}\over\sqrt{m(n+1-\theta)}}\right\}.\end{split} \] 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 the circle problem), the authors then prove Entry 1.1 given above.