Back
 AM  Vol.5 No.16 , September 2014
Asymptotic Harmonic Behavior in the Prime Number Distribution
Abstract: We consider on x > 0, where the sum is over all primes p. If Φ is bounded on x > 0, then the Riemann hypothesis is true or there are infinitely many zeros . The first 21 zeros give rise to asymptotic harmonic behavior in Φ(x) defined by the prime numbers up to one trillion.
Cite this paper: van Putten, M. (2014) Asymptotic Harmonic Behavior in the Prime Number Distribution. Applied Mathematics, 5, 2547-2557. doi: 10.4236/am.2014.516244.
References

[1]   Keller, H.B. (1987) Numerical Methods in Bifurcation Problems. Springer Verlag/Tata Institute for Fundamental Research, Berlin.

[2]   Hadamard, J. (1893) Etude sur les propriétés des fonctions entiéres et en particulier d’une fonction. Journal de Mathématiques Pures et Appliquées, 9, 171-216.

[3]   von Mangoldt, H. (1985) Zu Riemann’s Abhandlung “Über die Anzahl der Priemzahlen unter einer gegebenen Grösse”. Journal für die Reine und Angewandte Mathematik, 114, 255-305.

[4]   Titchmarsh, E.C. (1986) The Theory of the Riemann Zeta-Function. 2nd Edition, Oxford.

[5]   Lehmer, D.H. (1988) The Sum of Like Powers of the Zeros of the Riemann Zeta Function. Mathematics of Computation, 50, 265-273.
http://dx.doi.org/10.1090/S0025-5718-1988-0917834-X

[6]   Dusart, P. (1999) Inégalités explicites pour Ψ(X), θ(X), π(X) et les nombres premiers. Comptes Rendus Mathematiques (Mathematical Reports) des l’Academie des Sciences, 21, 53-59.

[7]   Keiper, J.B. (1992) Power Series Expansions of Riemann’s ζ Function. Mathematics of Computation, 58, 765-773.

[8]   Ford, K. (2002) Zero-Free Regions for the Riemann Zeta Function. Number Theory for the Millenium, 2, 25-26.

[9]   Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2006) The Riemann Hypothesis. Springer Verlag, Berlin.

[10]   Littlewood, J.E. (1922) Researches in the Theory of the Riemann ζ-Function. Proceedings of the London Mathematical Society, Series 2, 20, 22-27.

[11]   Littlewood, J.E. (1926) On the Riemann Zeta-Function. Proceedings of the London Mathematical Society, Series 2, 24, 175-201.
http://dx.doi.org/10.1112/plms/s2-24.1.175

[12]   Littlewood, J.E. (1928) Mathematical Notes (5): On the Function 1/ζ(1+ti). Proceedings of the London Mathematical Society, Series 2, 27, 349-357.
http://dx.doi.org/10.1112/plms/s2-27.1.349

[13]   Wintner, A. (1941) On the Asymptotic Behavior of the Riemann Zeta-Function on the Line . American Journal of Mathematics, 63, 575-580.
http://dx.doi.org/10.2307/2371370

[14]   Richert, H.E. (1967) Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1. Mathematische Annalen, 169, 97-101.
http://dx.doi.org/10.1007/BF01399533

[15]   Cheng, Y. (1999) An Explicit Upper Bound for the Riemann Zeta Function near the Line σ = 1. Rocky Mountain Journal of Mathematics, 29, 115-140.
http://dx.doi.org/10.1216/rmjm/1181071682

 
 
Top