TITLE:
Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros
AUTHORS:
Alfred Wünsche
KEYWORDS:
Riemann Zeta and Xi Function, Modified Bessel Functions, Second Mean-Value Theorem or Gauss-Bonnet Theorem, Riemann Hypothesis
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.3,
March
29,
2019
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.