On some trigonometric and exponential lattice sums (Q1342485)

The sums in question have the form \(\sum_{m,n} (\pm 1)^{m+n} r^{-1} \sin (r\theta)\) and \(\sum_{m,n} (\pm 1)^{m+n} r^{-1} \cos (r\theta)\), where \(r= \sqrt {m^ 2+ n^ 2}\), \(\theta\) is real, and the sums are extended over all nonzero lattice points \((m,n)\). These series do not converge absolutely and they may or may not converge conditionally. The authors show that the series are summable by a special Abelian method, and their Abelian sums are obtained. This is done by first evaluating an absolutely convergent exponential lattice series and then extending the sums by analytic continuation.