A new proof of the Vorono\"i summation formula
Mathematical Physics
2015-05-18 v4 math.MP
Abstract
We present a short alternative proof of the Vorono\"i summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schr\"odinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.
Keywords
Cite
@article{arxiv.1001.3556,
title = {A new proof of the Vorono\"i summation formula},
author = {Sebastian Egger and Frank Steiner},
journal= {arXiv preprint arXiv:1001.3556},
year = {2015}
}
Comments
Enlarged version of the published article J. Phys. A: Math. Theor. 44 (2011) 225302 (11pp)