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On an optimality property of Ramanujan sums

Authors:	Gennady Bachman

Institution:	Department of Mathematical Sciences, University of Nevada, Las Vegas, Nevada 89154-4020

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 , which reduces the summation over to a Ramanujan sum $c_{q}(a)=\sum _{\substack {n=1 (n,q)=1}}^{q}e^{2\pi ian/q}$ .

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