Accessibility navigation

'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

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


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 .

Item Type:Article
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:31486
Additional Information:10.7169/facm/2013.49.2.1
Publisher:Adam Mickiewicz University

University Staff: Request a correction | Centaur Editors: Update this record

Page navigation