On the Riemann Hypothesis in The Light of The Littlewood Criterion of Equivalence

Authors

  • Lukasz Andrzej Glinka Science Editor, BSc Physics. Full Member at the American Association of International Researchers, American Institute for Policy Development, New York

DOI:

https://doi.org/10.37591/rrdms.v1i2.135

Keywords:

Analytic number theory, Riemann zeta function, rie-mann hypothesis, Littlewood criterion, Mellin transforms, Mertens function

Abstract

In 1859, Riemann extended the Euler study onto the case of a complex variable according to then new Cauchy's complex analysis. According to his point of view, all non-trivial zeros of the Riemann zeta function are located on the critical line s=1/2+it, where t is a real number. In 1912, Littlewood presented the equivalence criterion for the Riemann Hypothesis based on the Mertens function. In this paper, the Riemann Hypothesis is discussed and this is shown that the Littlewood criterion can be immediately applied for proving the Riemann Hypothesis, according to the analytic number theory.

Author Biography

  • Lukasz Andrzej Glinka, Science Editor, BSc Physics. Full Member at the American Association of International Researchers, American Institute for Policy Development, New York

    BIRTH DATA 5 October 1984, Poland

    PROFESSIONAL WEB-PAGES

    http://www.linkedin.com/in/laglinka

    http://www.academicroom.com/users/laglinka

    http://www.goodreads.com/author/show/5832296

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    SUMMARY

    • Author. 2 science books and 1 book in humanities

    • Author. 42 science papers and manuscripts

    • Editor. Research & Reviews: Discrete Mathematical Structures (STM Journals, Consortium eLearning Network Private Ltd)

    • Editor. Review of Applied Physics (Science and Engineering Publishing Company)

    • Editor. Journal of Advances in Physics (Council for Innovative Research)

    • Editor. Science Journal of Applied Mathematics and Statistics (Science Publishing Group)

    • Editor. Applied Mathematics and Physics (Science and Education Publishing)

    • Attended 16 academic conferences, given 26 public presentations

    • Biography published by Marquis Who’s Who

    • Citations. More than 3 books and 75 papers of other authors

    EXPERIENCE

    • in 2013: IBM, Bratislava, Slovakia

    • in 2013: B.M. Birla Science Centre, Hyderabad, India

    • in 2011: TC Systems SA, Mendrisio, Switzerland

    • in 2010: Peoples’ Friendship University of Russia, Moscow, Russia

    • in 2009: B.M. Birla Science Centre, Hyderabad, India

    • in 2008: University of Udine, Udine, Italy

    • 2006-2008: Joint Institute for Nuclear Research, Dubna, Russia

    – in 2007: Centre de Physique Théorique de Marseille, Marseille, France

    – in 2007: University of Warsaw, Warsaw, Poland

    – in 2007: University of Wroclaw, Wroclaw, Poland

    – in 2006: Sołtan Institute for Nuclear Studies, Warsaw, Poland

    AUTHORED BOOKS

    L.A. Glinka (2013). Theorizing Emotions: A Brief Study of Psychological, Philosophical, and Cultural Aspects of Human Emotions. Cambridge International Science Publishing, 104 pages, ISBN-13: 978-1907343957. Information: www.cisp-publishing.com/acatalog/info_105.html

    L.A. Glinka (2013). Study of Analytic Number Theory: Riemann’s Hypothesis and Prime Number Theorem with Addendum on Integer Partitions. Cambridge International Science Publishing, 108 pages, ISBN-13: 978-1907343988. Information: www.cisp-publishing.com/acatalog/info_104.html

    L.A. Glinka (2012). Aethereal Multiverse: A New Unifying Theoretical Approach to Cosmology, Particle Physics, and Quantum Gravity. Cambridge International Science Publishing, 830 pages, ISBN-13: 978-1907343568. Information: www.cisp-publishing.com/acatalog/info_101.html

    CONTACT

    e-mail: [email protected]

    mobile phone: (+48) 660 019 237

References

Euler L., Comm. Acad. Sci. Petropol. 5; 36 (1730), 9; 222 (1737), 9; 160 (1744); Introduction in Analysin In nitorum (M.-M. Bousquet & Socios, 1748); In-stitutiones Calculi Dierentialis Cum Eius Usu in Analysi Finitorum Ac Doctrina Serierum (Acad. Imp. Sci. Petropol.; Mich lis, 1755); Mem. Acad. Sci. Berlin 17, 83 (1768); Novi Comm. Acad. Sci. Petropol. 4, 105 (1771).

Riemann B., Monats. Kgl. Preuss. Akad. Wiss. Berlin, 671 (1859). Also in R. Dedekind and H. Weber (Eds.), Bernhard Riemann's Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, 145–155p (2nd ed., Teubner, 1892).

Cauchy A.L., Memoire sur les integrales de nies prises entre des lim-ites imaginaires (De Bure Freres, 1825); Sur un nouveau genre de calcul analogue au calcul in nitesimall, in A.L. Cauchy, Exercices de Mathe-matiques, pp. 11-28 (De Bure Freres, 1826); Sur la mecanique ceeste et sur un nouveau calcul qui s'applique a un grande nombre de questions diverses, in De M. le Baron de Ferussac (Ed.), Bulletin des Sciences Mathematiques, Physiques et Chimiques, Tome XV, pp. 260-276 (Firmin Didot, 1831); Memoire sur les rapports qui existent entre le calcul des Residus et le calcul des Limites, et sur les avantages qu'o rent ces deux calculs dans la resolution des equations algebriques ou transcendantes, in De M. le Baron de Ferussac (Ed.), Bulletin Des Sciences Mathema-tiques, Physiques et Chimiques, Tome XVI, pp. 116-133 (Firmin Didot, 1831).

Hilbert D., Nachr. Kgl. Ges. Wiss. Gottingen 3, 253 (1900); Arch. f. Math. u. Phys. 3, 44 & 213 (1901); Bull. AMS 8(10), 437 (1902); in E. Duporcq (Ed.), Compte Rendu du Deuxieme Congres International de Mathematiciens, 58–114p. (Gauthiers-Villars, 1902).

Smale S., Mathematical problems for the next century, in V. Arnold, M. Atiyah, P. Lax, and B. Mazur (Eds.) Mathematics: Frontiers and Perspectives, 271–294p (AMS, 2000).

Bombieri E., The Riemann Hypothesis, O cial Problem Description (Clay Mathematics Institute, 2000); Sarnak, Problems of the Millennium: The Riemann Hypothesis (2004) (Clay Mathematics Institute, 2004).

Stieltjes T.J., Lettre a Hermite de 11 Juillet 1885: Lettre 79, in B. Baillaud and H. Bourget (Eds.), Prefaced by E. Picard, Correspondance d'Hermite et de Stieltjes. Tome I (8 novembre 1882 - 22 juillet 1889) (Gauthier-Villars, 1905);

Mertens F., Sitz. d. Kaiser. Akad. d. Wiss. Mat.-Nat. Kl. 1897 Ab. 2A 106, 761p.

Hardy G.H., Divergent Series (Oxford University Press, 1949); K. Knopp, In nite Sequences and Series (Dover, 1956); G.M. Fichtenholz, In nite Series: Rami cations (Gordon & Breach, 1970);W.F. Trench, Introduction to Real Analysis (Prentice Hall, 2002); D.D. Bonar and M.J. Khoury, Real In nite Series (MAA 2006).

Apostol T.M., Modular Functions and Dirichlet Series in Number Theory, 2nd Edn.: Springer, 1990.

Littlewood J.E., C. R. Acad. Sci. Paris 1912; 154:263p.

Glinka L.A., Study of Analytic Number Theory: Riemann's Hypothe-sis and Prime Number Theorem with Addendum on Integer Partitions; Cambridge International Science Publishing 2013.

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Published

2014-09-08

Issue

Vol. 1 No. 2 (2014)

Section

Research Articles

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