On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform
Abstract
:1. Introduction
2. Estimate for a Metric
3. Limit Lemma
4. Proof of Theorem 2
5. Identification of Set
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- The Millennium Prize Problems. Available online: https://www.claymath.org/millennium-problems/ (accessed on 20 June 2023).
- Elizalde, E. Zeta-functions: Formulas and applications. J. Comput. Appl. Math. 2000, 118, 125–142. [Google Scholar] [CrossRef] [Green Version]
- Elizalde, E. Zeta-functions and the cosmos—A basic brief review. Universe 2021, 7, 5. [Google Scholar] [CrossRef]
- Maino, G. Prime numbers, atomic nuclei, symmetries and superconductivity. AIP Conf. Proc. 2019, 2150, 030009. [Google Scholar]
- Aref’eva, I.Y.; Volovich, I.V. Quantization of the Riemann zeta-function and cosmology. Int. J. Geom. Methods Mod. Phys. 2007, 4, 881–895. [Google Scholar] [CrossRef] [Green Version]
- Gutzwiller, M.C. Stochastic behavior in quantum scattering. Physica 1983, 7D, 341–355. [Google Scholar] [CrossRef]
- Voronin, S.M. Theorem on the “universality” of the Riemann zeta-function. Math. USSR Izv. 1975, 9, 443–453. [Google Scholar] [CrossRef]
- Motohashi, Y. A relation between the Riemann zeta-function and the hyperbolic Laplacian. Ann. Sc. Norm. Super. Pisa-Cl. Sci. IV Ser. 1995, 22, 299–313. [Google Scholar]
- Ivić, A. On some conjectures and results for the Riemann zeta-function and Hecke series. Acta Arith. 2001, 99, 115–145. [Google Scholar] [CrossRef] [Green Version]
- Ivić, A. The Mellin transform and the Riemann zeta-function. In Proceedings of the Conference on Elementary and Analytic Number Theory, Vienna, Austria, 18–20 July 1996; Nowak, W.G., Schoißengeier, J., Eds.; Universität Wien & Universität für Bodenkultur: Vienna, Austria, 1996; pp. 121–127. [Google Scholar]
- Ivić, A. On the error term for the fourth moment of the Riemann zeta-function. J. Lond. Math. Soc. 1999, 60, 21–32. [Google Scholar]
- Ivić, A.; Motohashi, Y. The mean square of the error term for the fourth moment of the zeta-function. Proc. Lond. Math. Soc. 1994, 69, 309–329. [Google Scholar] [CrossRef]
- Ivić, A.; Motohashi, Y. The fourth moment of the Riemann zeta-function. J. Number Theory 1995, 51, 16–45. [Google Scholar]
- Jutila, M. The Mellin transform of the square of Riemann’s zeta-function. Period. Math. Hung. 2001, 42, 179–190. [Google Scholar] [CrossRef]
- Lukkarinen, M. The Mellin Transform of the Square of Riemann’s Zeta-Function and Atkinson Formula. Ph.D. Thesis, University of Turku, Turku, Finland, 2004. [Google Scholar]
- Ivić, A.; Jutila, M.; Motohashi, Y. The Mellin transform of powers of the zeta-function. Acta Arith. 2000, 95, 305–342. [Google Scholar] [CrossRef] [Green Version]
- Korolev, M.; Laurinčikas, A. On the approximation by Mellin transform of the Riemann zeta-function. Axioms 2023, 12, 520. [Google Scholar]
- Billingsley, P. Convergence of Probability Measures; Wiley: New York, NY, USA, 1968. [Google Scholar]
- Laurinčikas, A. Limit Theorems for the Riemann Zeta-Function; Kluwer: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 1996. [Google Scholar]
- Steuding, J. Value-Distribution of L-Functions; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2007; Volume 1877. [Google Scholar]
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Laurinčikas, A. On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform. Axioms 2023, 12, 789. https://doi.org/10.3390/axioms12080789
Laurinčikas A. On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform. Axioms. 2023; 12(8):789. https://doi.org/10.3390/axioms12080789
Chicago/Turabian StyleLaurinčikas, Antanas. 2023. "On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform" Axioms 12, no. 8: 789. https://doi.org/10.3390/axioms12080789