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

Hilberdink, T. (2013) 'Quasi'-norm of an arithmetical convolution operator and the order of the Riemann zeta function. Functiones et Approximatio: Commentarii Mathematici, 49 (2). pp. 201-220. ISSN 0208-6573 doi: 10.7169/facm/2013.49.2.1

Abstract/Summary

In this paper we study Dirichlet convolution with a given arithmetical function f as a linear mapping 'f that sends a sequence (an) to (bn) where bn = Pdjn f(d)an=d. We investigate when this is a bounded operator on l2 and ¯nd the operator norm. Of particular interest is the case f(n) = n¡® for its connection to the Riemann zeta function on the line <s = ®. For ® > 1, 'f is bounded with k'f k = ³(®). For the unbounded case, we show that 'f : M2 ! M2 where M2 is the subset of l2 of multiplicative sequences, for many f 2 M2. Consequently, we study the `quasi'-norm sup kak = T a 2M2 k'fak kak for large T, which measures the `size' of 'f on M2. For the f(n) = n¡® case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of j³(® + iT )j for ® > 12 .