'Quasi'-norm of an arithmetical convolution operator and the order of the Riemann zeta functionHilberdink, 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 Full text not archived in this repository. It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.7169/facm/2013.49.2.1 Abstract/SummaryIn 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 .
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