An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function

An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function

Given a periodic function f, we study the convergence almost everywhere and in norm of the series ∑kckf(kx). Let f(x)=∑m=1∞amsin⁡2πmx where ∑m=1∞am2d(m)1/2, by only using elementary Dirichlet convolution calculus, we show that for 0

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Veröffentlicht in:Journal of number theory 2016-05, Vol.162, p.137-179
1. Verfasser: Weber, Michel J.G.
Format: Artikel
Sprache:eng
Schlagworte:
Almost everywhere convergence
Arithmetical functions
Decomposition of squared sums
Dirichlet convolution
FC sets
GCD
Mathematics
Mean convergence
Number Theory
Riemann Zeta function
Series
Systems of dilated functions
Ω-theorem
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Beschreibung
Zusammenfassung:Given a periodic function f, we study the convergence almost everywhere and in norm of the series ∑kckf(kx). Let f(x)=∑m=1∞amsin⁡2πmx where ∑m=1∞am2d(m)1/2, by only using elementary Dirichlet convolution calculus, we show that for 0
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2015.10.002