Abstract
We prove that the Kloosterman sum \(S(1,1;c)\) changes sign infinitely often as \(c\) runs over squarefree moduli with at most 10 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler and Matomäki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.
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Acknowledgments
The present work in this paper will be part of my PhD thesis. I am grateful to Professor Philippe Michel for his kind supervision and suggesting this problem to me. His valuable suggestions and comments should be greatly acknowledged. I also thank Paul Nelson for his helpful comments on sieve theory and everything else. The idea in this paper was partially inspired by a talk of Professor Kai-Man Tsang during a conference celebrating the 25 years Number Theory Seminar at ETH Zürich in June 2013. I would like to thank Professor Tsang and the organizers of the conference. Sincere thanks are also due to Professor Yuan Yi for her constant help and encouragement. The numerical computations in this paper are based on the Mathematica codes of Kaisa Matomäki, and I thank her for sharing the codes on her homepage. I am also grateful to the referee for his/her detailed comments and suggestions, which have greatly improved the exposition of the paper.
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Communicated by J. Schoißengeier .
The work is partially supported by China Scholarship Council and N.S.F. (No. 11171265) of P. R. China.
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Xi, P. Sign changes of Kloosterman sums with almost prime moduli. Monatsh Math 177, 141–163 (2015). https://doi.org/10.1007/s00605-014-0653-z
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DOI: https://doi.org/10.1007/s00605-014-0653-z