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Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros
Institut für Physik, Humboldt-Universität, MPG Nichtklassische Strahlung, Berlin, Germany
- 1 Institut für Physik, Humboldt-Universität, MPG Nichtklassische Strahlung, Berlin, Germany
Advances in Pure Mathematics·Volume 09 (2019)·Pages 281–316·Published 29 March 2019·DOI10.4236/apm.2019.93013
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Abstract
The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.
KeywordsRiemann Zeta and Xi FunctionModified Bessel FunctionsSecond Mean-Value Theorem or Gauss-Bonnet TheoremRiemann Hypothesis
- Riemann, B. (1859) über die Anzahl der Primzahlen unter einer gegebenen Grosse, Monatsber. Akad, Berlin, 671-680.
- Edwards, H.M. (1974) Riemann’s Zeta Function. Dover, New York.
- Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2008) The Riemann Hypothesis. Springer, New York.
- Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis. Cambridge University Press, Cambridge.
- Titchmarsh, E.C. (1951) The Theory of the Riemann Zeta-Function. Oxford University Press, Oxford.
- Chandrasekharan, K. (1970) Arithmetical Functions. Springer-Verlag, Berlin.
- Ivic, A. (1985) The Riemann Zeta-Function. Dover Publications, Mineola.
- Apostol, M. (2018) Introduction to Analytic Number Theory. Springer, New York.
- Neuberger, J.W., Feiler, C., Maier, H. and Schleich, W.P. (2014) Newton Flow of the Riemann Zeta Function: Separatrices Control the Appearance of Zeros. New Journal of Physics, 16, Article ID: 103023. https://doi.org/10.1088/1367-2630/16/10/103023
- Neuberger, J.W., Feiler, C., Maier, H. and Schleich, W.P. (2015) The Riemann Hypothesis Illuminated by the Newton Flow of ζ. Physica Scripta, 90, Article ID: 108015.
- Wünsche, A. (2016) Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus, General Mathematics. Advances in Pure Mathematics, 6, 972-1021. https://doi.org/10.4236/apm.2016.613074
- Watson, G.N. (1944) A Treatise on the Theory of Bessel Functions. 2nd Edition, Cambridge University Press, Cambridge.
- Korenev, B.G. (2002) Introduction into the Heory of Bessel Functions. Taylor and Francis, Oxford.
- Courant, R. (1937) Differential and Integral Calculus. Vol. 1, 2nd Edition, Interscience, New York.
- Widder, D.V. (1989) Advanced Calculus. 2nd Edition, Dover Publications Inc., New York.
- Katsnelson, V. Email from 06.01.2017.
- Wünsche, A. (2017) Correction and Supplement to Approach for a Proof of Riemann Hypothesis by Second Mean-Value Theorem. Advances in Pure Mathematics, 7, 263-276. https://doi.org/10.4236/apm.2017.73013
- Gélinas, J. (2018) Refutation of a Proof of the Riemann Hypothesis by the Bonnet Mean-Value Theorem. Unpublished.