A new proof of the Vorono\"i summation formula

TL;DR

This paper offers a concise alternative proof of the Vorono"i summation formula, highlighting its significance in number theory and physics, particularly in quantum graph trace formulas and Lambert series identities.

Contribution

It provides a novel, simplified proof of the Vorono"i summation formula and establishes a new proof of a Lambert series identity related to the divisor function.

Findings

New proof of the Vorono"i summation formula

Derivation of a Lambert series identity involving divisor function

Connection between the Lambert series and quantum graph trace

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}.