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Title: An Alternative Form of the Functional Equation for Riemann’s Zeta Function, II (English)
Author: Ossicini, Andrea
Language: English
Journal: Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
ISSN: 0231-9721
Volume: 53
Issue: 2
Year: 2014
Pages: 115-138
Summary lang: English
Category: math
Summary: This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for $\zeta (s)$. We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau rapport entre les séries des puissances tant direct que réciproches”. This more general functional equation gives origin to a special function,here named $(s)$ which we prove that it can be continued analytically to an entire function over the whole complex plane using techniques similar to those of the second proof of Riemann. Moreover we are able to obtain a connection between Jacobi’s imaginary transformation and an infinite series identity of Ramanujan. Finally, after studying the analytical properties of the function $(s)$, we complete and extend the proof of a Fundamental Theorem, both on the zeros of Riemann Zeta function and on the zeros of Dirichlet Beta function, using also the Euler–Boole summation formula. (English)
Keyword: Riemann Zeta
Keyword: Dirichlet Beta
Keyword: generalized Riemann hypothesis
Keyword: series representations
MSC: 11B68
MSC: 11M06
MSC: 11M26
idZBL: Zbl 1308.11078
idMR: MR3331010
Date available: 2014-12-16T15:11:16Z
Last updated: 2020-01-05
Stable URL:
Reference: [1] Backlund, R.: Sur les zéros de la fonction $\zeta (s)$ de Riemann. C. R. Acad. Sci. Paris 158 (1914), 1979–1982.
Reference: [2] Bellman, R. A.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, New York, 1961. Zbl 0098.28301, MR 0125252
Reference: [3] Berndt, B. C.: Ramanujan’s Notebooks. Part II. Springer-Verlag, New York, 1989. Zbl 0716.11001, MR 0970033
Reference: [4] Borwein, J. M., Calkin, N. J., Manna, D.: Euler-Boole summation revisited. American Mathematical Monthly 116, 5 (2009), 387–412. Zbl 1229.11035, MR 2510837, 10.4169/193009709X470290
Reference: [5] Ditkine, V., Proudnikov, A.: Transformations Integrales et Calcul Opèrationnel. Mir, Moscow, 1978. MR 0622210
Reference: [6] Edwards, H. M.: Riemann’s Zeta function. Pure and Applied Mathematics 58, Academic Press, New York–London, 1974. Zbl 0315.10035, MR 0466039
Reference: [7] Erdelyi, I. et al.: Higher Trascendental Functions. Bateman Manuscript Project 1, McGraw-Hill, New York, 1953.
Reference: [8] Euler, L.: Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques. Hist. Acad. Roy. Sci. Belles-Lettres Berlin 17 (1768), 83–106, (Also in: Opera Omnia, Ser. 1, vol. 15, 70–90).
Reference: [9] Finch, S. R.: Mathematical Constants. Cambridge Univ. Press, Cambridge, 2003. Zbl 1054.00001, MR 2003519
Reference: [10] Ingham, A. E.: The Distribution of Prime Numbers. Cambridge Univ. Press, Cambridge, 1990. Zbl 0715.11045, MR 1074573
Reference: [11] Jacobi, C. G. I.: Fundamenta Nova Theoriae Functionum Ellipticarum. Sec. 40, Königsberg, 1829.
Reference: [12] Lapidus, M. L., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimension and Zeta Functions. Springer-Verlag, New York, 2006. MR 2245559
Reference: [13] Legendre, A. M.: Mémoires de la classe des sciences mathématiques et phisiques de l’Institut de France, Paris. (1809), 477–490.
Reference: [14] Ossicini, A.: An alternative form of the functional equation for Riemann’s Zeta function. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008/9), 95–111. MR 2604733
Reference: [15] Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Gesammelte Werke, Teubner, Leipzig, 1892, reprinted Dover, New York, 1953, first published Monatsberichte der Berliner Akademie, November 1859.
Reference: [16] Stirling, J.: Methodus differentialis: sive tractatus de summatione et interpolatione serierum infinitarum. Gul. Bowyer, London, 1730.
Reference: [17] Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht–Boston–London, 2001. Zbl 1014.33001, MR 1849375
Reference: [18] Titchmarsh, E. C., Heath-Brown, D. R.: The Theory of the Riemann Zeta-Function. 2nd ed., Oxford Univ. Press, Oxford, 1986. MR 0882550
Reference: [19] Varadarajan, V. S.: Euler Through Time: A New Look at Old Themes. American Mathematical Society, 2006. Zbl 1096.01013, MR 2219954
Reference: [20] Varadarajan, V. S.: Euler and his work of infinite series. Bulletin of the American Mathematical Society 44, 4 (2007), 515–539. MR 2338363, 10.1090/S0273-0979-07-01175-5
Reference: [21] Weil, A.: Number Theory: an Approach Through History from Hammurapi to Legendre. Birkhäuser, Boston, 2007. Zbl 1149.01013, MR 2303999
Reference: [22] Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis. 4th ed., Cambridge Univ. Press, Cambridge, 1988. MR 1424469


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