Abstract
The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.
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Acknowledgments
This work is based on Chapter 1 of the author’s PhD thesis [14]. I would like to sincerely thank my advisors Samuel J. Patterson and Philippe Michel for their encouragements and support. Part of this work has been realized at the University Montpellier 2 and at the Swiss Federal Institute of Technology. I would also like to thank Valentin Blomer for interesting suggestions and advice, and the anonymous referee for his very careful reading of the manuscript.
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Communicated by J. Schoißengeier.
Author supported by the Volkswagen Foundation.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Louvel, B. The first moment of Salié sums. Monatsh Math 168, 523–543 (2012). https://doi.org/10.1007/s00605-011-0366-5
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DOI: https://doi.org/10.1007/s00605-011-0366-5