On the convergence of a two-dimensional trigonometric sum (Q1353719)

Starting point of this paper are considerations on the convergence of the two-dimensional trigonometric sums \[ S(\nu)= \sum^\infty_{m=1} \sum^\infty_{n= 0}(-1)^{m+ n} {\sin(2\nu\sqrt{m^2+ n^2})\over \sqrt{m^2+ n^2}} \] and \[ C(\nu)= \sum^\infty_{m=1} \sum^\infty_{n= 0}(-1)^{m+ n} {\cos(2\nu\sqrt{m^2+ n^2})\over \sqrt{m^2+ n^2}}. \] The author investigates more generally the behaviour of \[ E_{t,u}(x)= \sum_{1\leq n\leq x} \sum_{k\in \mathbb{Z}^2} {e^{2\pi i\langle u,k\rangle+ 2\pi\sqrt{tn}}\over \sqrt n} \] for real \(t\geq 0\), \(x\geq 1\) and \(u\in \mathbb{R}^2\). Here \(\langle.,.\rangle\) denotes the standard scalar product. For \(\nu\geq 0\) the partial sums of \(S(\nu)\) and \(C(\nu)\) (in the sense of spherical summation) are the imaginary and real part of \(4^{-1}E_{t,u}(x)\) with \(t= \nu^2/\pi^2\) and \(u=\left({1\over 2},{1\over 2}\right)\).