Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

On Prime Numbers and The Riemann Zeros

Download PDF (869.8 KB) PP. 208 - 219 Pub. Date: October 24, 2017

DOI: 10.22606/jaam.2017.24002

Author(s)

  • Lucian M. Ionescu*
    Department of Mathematics, Illinois State University, IL 61790-4520, United States

Abstract

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros [1,2,3]. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: adelic duality and the POSet of prime numbers. The article presents computational evidence of the structure of the imaginary parts t of the non-trivial zeros of the Riemann zeta function ρ = 1/2 + it, called in this article the Riemann Spectrum, using the study of their distribution, as in [3]. The novelty represents in considering the associated characters pit, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied in [4], is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and !-eigen-distributions, are explored following [5].

Keywords

Riemann zeta function, prime numbers, adeles, duality, distributions

References

[1] V. Munoz and R. P. Marco, “Unified treatment of explicit and trace formulas via Poisson-Newton formula”, https://arxiv.org/abs/1309.1449

[2] O. Shanker, “Entropy of Riemann zeta zero sequence”, AMO - Advanced Modeling and Optimization, Volume 15, Number 2, 2013, https://camo.ici.ro/journal/vol15/v15b18.pdf

[3] K. Ford and A. Zaharescu, “On the distribution of imaginary parts” of zeros of the Riemann zeta function, J. reine angew. Math. 579 (2005), 145-158. www.researchgate.net, 2005.

[4] L. M. Ionescu, “A natural partial order on the prime numbers”, Notes on Number Theory and Discrete Mathematics, Volume 21, 2015, Number 1, Pages 1?9; arxiv.org/abs/1407.6659, 2014.

[5] S. S. Kudla, “Tate’s thesis”, An Introduction to the Langlands Program, pp 109-131, Editors: Joseph Bernstein, Stephen Gelbart, Springer, 2004.

[6] L.M. Ionescu, “Remarks on physics as number theory”, Proceedings of the 19th National Philosophy Alliance Vol. 9, pp. 232-244, http://www.gsjournal.net/old/files/4606_Ionescu2.pdf.

[7] B. Mazur, W. Stein, Primes: What is Riemann’s Hypothesis?, Cambridge University Press, 2016; http://modular.math.washington.edu/rh/rh.pdf

[8] V. R. Pratt, “Every prime has a succinct certificate”, SIAM J. Comput. Vol.4, No.3, Sept. 1975, 214-220.

[9] K. Ford, K. Soundararajan, A. Zaharescu, “On the distribution of imaginary parts of zeros of the Riemann zeta function, II”, 0805.2745, 2009.

[10] H. Rademacher, Fourier analysis in number theory, Collected Works, pp.434-458.

[11] K. Conrad, “The character group of Q”, http://www.math.uconn.edu/?kconrad/blurbs/gradnumthy/characterQ.pdf

[12] I. M. Gel’fand M. I. Graev, I. I. Pyateskii-Shapiro, Representation theory and automorphic forms, Academic Press, 1990.

[13] P. Garrett, “Riemann’s explicit/exact formula”, 2015, http://www-users.math.umn.edu/?garrett/m/mfms/notes_2015-16/03_Riemann_and_zeta.pdf

[14] F. Gouvea, p-adic Numbers: An Introduction, Springer-Verlag, 1993.

[15] L. M. Ionescu, Recent presentations, http://my.ilstu.edu/ lmiones/presentations_drafts.htm

[16] R. Meyer, “A spectral interpretation of the zeros of the Riemann zeta function”, math/0412277

[17] J. F. Burnol, “Spectral analysis of the local conductor operator”, math/9809119.pdf, 1998.

[18] A. Connes, “An essay on the Riemann Hypothesis”, http://www.alainconnes.org/docs/rhfinal.pdf

[19] John Baez, “Quasicrystals and the Riemann Hypothesis”, golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html

[20] Freeman Dyson, “Frogs and Birds”, AMS 56 (2009), 212-223.

[21] A. M. Odlyzko, “Primes, quantum chaos and computers”, Proc. Symp., May 1989, Washington DC, pp.35-46.

[22] Wikipedia: “Riemann Zeta Function”.

[23] L. M. Ionescu, “A statistics study of Riemann zeros”, 2014, to appear.