On an optimality property of Ramanujan sums

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 $c_{q}(a)=\sum _{\substack {n=1 (n,q)=1}}^{q}e^{2\pi ian/q}$.