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An alternative form of the functional equation for Riemann’s zeta function. II. (English) Zbl 1308.11078

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 \(\text Ѐ(s)\) 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 \(\text Ѐ(s)\), 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.
For Part I see [Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56(2008–2009), 95–111 (2009; Zbl 1219.11124)].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 1219.11124
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References:

[1] Backlund, R.: Sur les zéros de la fonction \(\zeta (s)\) de Riemann. C. R. Acad. Sci. Paris 158 (1914), 1979-1982. · JFM 45.0719.02
[2] Bellman, R. A.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, New York, 1961. · Zbl 0098.28301
[3] Berndt, B. C.: Ramanujan’s Notebooks. Part II. Springer-Verlag, New York, 1989. · Zbl 0716.11001
[4] Borwein, J. M., Calkin, N. J., Manna, D.: Euler-Boole summation revisited. American Mathematical Monthly 116, 5 (2009), 387-412. · Zbl 1229.11035
[5] Ditkine, V., Proudnikov, A.: Transformations Integrales et Calcul Opèrationnel. Mir, Moscow, 1978. · Zbl 0494.44001
[6] Edwards, H. M.: Riemann’s Zeta function. Pure and Applied Mathematics 58, Academic Press, New York-London, 1974. · Zbl 0315.10035
[7] Erdelyi, I. et al.: Higher Trascendental Functions. Bateman Manuscript Project 1, McGraw-Hill, New York, 1953.
[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)
[9] Finch, S. R.: Mathematical Constants. Cambridge Univ. Press, Cambridge, 2003. · Zbl 1054.00001
[10] Ingham, A. E.: The Distribution of Prime Numbers. Cambridge Univ. Press, Cambridge, 1990. · Zbl 0715.11045
[11] Jacobi, C. G. I.: Fundamenta Nova Theoriae Functionum Ellipticarum. Sec. 40, Königsberg, 1829.
[12] Lapidus, M. L., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimension and Zeta Functions. Springer-Verlag, New York, 2006. · Zbl 1119.28005
[13] Legendre, A. M.: Mémoires de la classe des sciences mathématiques et phisiques de l’Institut de France, Paris. (1809), 477-490.
[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. · Zbl 1219.11124
[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.
[16] Stirling, J.: Methodus differentialis: sive tractatus de summatione et interpolatione serierum infinitarum. Gul. Bowyer, London, 1730.
[17] Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht-Boston-London, 2001. · Zbl 1014.33001
[18] Titchmarsh, E. C., Heath-Brown, D. R.: The Theory of the Riemann Zeta-Function. 2nd, Oxford Univ. Press, Oxford, 1986.
[19] Varadarajan, V. S.: Euler Through Time: A New Look at Old Themes. American Mathematical Society, 2006. · Zbl 1096.01013
[20] Varadarajan, V. S.: Euler and his work of infinite series. Bulletin of the American Mathematical Society 44, 4 (2007), 515-539. · Zbl 1135.01010
[21] Weil, A.: Number Theory: an Approach Through History from Hammurapi to Legendre. Birkhäuser, Boston, 2007. · Zbl 1149.01013
[22] Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis. 4th, Cambridge Univ. Press, Cambridge, 1988. · Zbl 0108.26903
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