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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Exact estimates for integrals involving
Dirichlet series with nonnegative coefficients


Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 127 (1999), 2417-2422
MSC (1991): Primary 30B50; Secondary 30B10
Published electronically: April 9, 1999
MathSciNet review: 1600125
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Abstract: We consider the Dirichlet series

\begin{displaymath}\sum^\infty _{k=2} a_k k^{-1-x} =: f(x), \quad x>0, \end{displaymath}

with coefficients $a_k \ge 0$ for all $k$. Among others, we prove exact estimates of certain weighted $L^p$-norms of $f$ on the unit interval $(0,1)$ for any $0<p<\infty$, in terms of the coefficients $a_k$. Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series

\begin{displaymath}\sum^\infty _{k=1} c_k k^{-z}, \quad z:= x+iy, \end{displaymath}

with complex coefficients $c_k$.


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Additional Information

Ferenc Móricz
Affiliation: Bolyai Institute, University of Szeged, Aradi Vertanuk Tere 1, 6720 Szeged, Hungary
Email: moricz@math.u-szeged.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04851-0
PII: S 0002-9939(99)04851-0
Keywords: Dirichlet series, power series, Cauchy-Hadamard criterion for Dirichlet series, line of convergence, $L^p$-behavior, weight function, slowly decreasing function, Cauchy condensation principle
Received by editor(s): November 19, 1997
Published electronically: April 9, 1999
Additional Notes: This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 016 393.
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society