``Quasi''-norm of an arithmetical convolution operator and the order of the Riemann zeta function

Abstract

In this paper we study Dirichlet convolution with a given arithmetical function $f$ as a~linear mapping $\varphi_f$ that sends a sequence $(a_n)$ to $(b_n)$ where $b_n=\sum_{d|n} f(d)a_{n/d}$. We investigate when this is a bounded operator on $l^2$ and find the operator norm. Of particular interest is the case $f(n)=n^{-\alpha}$ for its connection to the Riemann zeta function on the line $\Re s =\alpha$. For $\alpha>1$, $\varphi_f$ is bounded with $\|\varphi_f\|=\zeta(\alpha)$. For the unbounded case, we show that $\varphi_f:\mathcal{M}^2\to\mathcal{M}^2$ where $\mathcal{M}^2$ is the subset of $l^2$ of multiplicative sequences, for many $f\in\mathcal{M}^2$. Consequently, we study the `quasi'-norm \[ \sup_{\substack{\|a\|=T\\ a\in\mathcal{M}^2} } \frac{\|\varphi_f a\|}{\|a\|} \] for large $T$, which measures the `size' of $\varphi_f$ on $\mathcal{M}^2$. For the $f(n)=n^{-\alpha}$ case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of $|\zeta (\alpha+iT)|$ for $\alpha>\frac{1}{2}$.