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arXiv · 1001.3556

A new proof of the Vorono\"i summation formula

Sebastian Egger·Frank Steiner
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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}.

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Sebastian Egger, Frank Steiner. 2010-01-20. A new proof of the Vorono\"i summation formula. https://doi.org/10.1088/1751-8113/44/22/225302

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