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Proceedings of the Royal Society of Edinburgh Section A: Mathematics Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Article contents

Explicit asymptotics for certain single and double exponential sums

Part of: Exponential sums and character sums Miscellaneous topics of analysis in the complex domain Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  22 January 2019

K. Kalimeris  and
A. S. Fokas
K. Kalimeris
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, UK (kk364@cam.ac.uk)
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, UK and Viterbi School of Engineering, University of Southern California, Los Angeles, CA90089-2560, USA
Corresponding
E-mail address:
kk364@cam.ac.uk
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Abstract

By combining classical techniques together with two novel asymptotic identities derived in recent work by Lenells and one of the authors, we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of Re{u} and Re{v} and for a variety of sets of summation, as well as particular cases of Mordell-Tornheim double sums.

Keywords

asymptotic expansions exponential sums

MSC classification

Primary: 11L07: Estimates on exponential sums
Secondary: 30E15: Asymptotic representations in the complex domain 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 607 - 632
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

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4Ishikawa, H. and Matsumoto, K.. On the estimation of the order of Euler-Zagier multiple zeta-functions. Illinois J. Math. 47 (2003), 1151–1166.CrossRefGoogle Scholar
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