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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On an optimality property of Ramanujan sums


Author: Gennady Bachman
Journal: Proc. Amer. Math. Soc. 125 (1997), 1001-1003
MSC (1991): Primary 11L03
DOI: https://doi.org/10.1090/S0002-9939-97-03650-2
MathSciNet review: 1363445
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Abstract: We evaluate $\inf _{b_{n}}\sum _{a=1}^{q}|\sum _{\substack {n=1\\ (n,q)=1}}^{q} b_{n}e^{2\pi ian/q}|$, where the $\inf $ is taken over sequences $b_{n}$ satisfying $b_{n}\ge 1$. In particular we show that it is attained by taking $b_{n}=1$ for all $n$, which reduces the summation over $n$ to a Ramanujan sum $c_{q}(a)=\sum _{\substack {n=1\\ (n,q)=1}}^{q}e^{2\pi ian/q}$.


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Additional Information

Gennady Bachman
Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, Nevada 89154-4020
Email: bachman@nevada.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03650-2
Received by editor(s): March 28, 1995
Received by editor(s) in revised form: October 19, 1995
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society