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# Title: Convergence of series of dilated functions and spectral norms of GCD matrices

Abstract: We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-\alpha})$ for some $\alpha \in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.
 Comments: 26 pages. Version 2: some minor changes (typos etc.). The manuscript has been accepted for publication by Acta Arithmetica Subjects: Classical Analysis and ODEs (math.CA); Number Theory (math.NT); Probability (math.PR); Spectral Theory (math.SP) MSC classes: 42A16, 42A20, 42A61, 42B05, 11A05, 15A18, 26A45 Cite as: arXiv:1407.5403 [math.CA] (or arXiv:1407.5403v2 [math.CA] for this version)

## Submission history

From: Christoph Aistleitner [view email]
[v1] Mon, 21 Jul 2014 07:51:45 GMT (26kb)
[v2] Tue, 2 Dec 2014 08:59:35 GMT (26kb)

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