Exact estimates for integrals involving Dirichlet series with nonnegative coefficients
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- by Ferenc Móricz PDF
- Proc. Amer. Math. Soc. 127 (1999), 2417-2422 Request permission
Abstract:
We consider the Dirichlet series \[ \sum ^\infty _{k=2} a_k k^{-1-x} =: f(x), \quad x>0, \] 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 \[ \sum ^\infty _{k=1} c_k k^{-z}, \quad z:= x+iy, \] with complex coefficients $c_k$.References
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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
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2417-2422
- MSC (1991): Primary 30B50; Secondary 30B10
- DOI: https://doi.org/10.1090/S0002-9939-99-04851-0
- MathSciNet review: 1600125