Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros - Advances in Pure Mathematics

APM > Vol.9 No.3, March 2019

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros ()

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Download as PDF (Size: 8637KB) PP. 281-316

DOI: 10.4236/apm.2019.93013 1,178 Downloads 2,887 Views Citations

Author(s)

Alfred Wünsche

Affiliation(s)

Institut für Physik, Humboldt-Universität, MPG Nichtklassische Strahlung, Berlin, Germany.

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.

KEYWORDS

Riemann Zeta and Xi Function, Modified Bessel Functions, Second Mean-Value Theorem or Gauss-Bonnet Theorem, Riemann Hypothesis

Share and Cite:

Wünsche, A. (2019) Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros. Advances in Pure Mathematics, 9, 281-316. doi: 10.4236/apm.2019.93013.

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