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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.
Highlights
The present paper tries to find out the common ground for the zeros of the Riemann zeta function ζ = ( z) ζ ( x + iy) and of the modified Bessel functionsIν ( z) (or Bessel functions Jν ( z ) of imaginary argument z) for imaginary argument z and, for the absence of zeros of the Gaussian Bell ( ) function exp z2
In present paper we will mainly have to do only with this Xi function ξ ( z) which we displaced in a way that its zeros lie on the imaginary axis provided; the Riemann Hypothesis is correct and we denote as Xi function
In this article we illustrated the behavior of the zeros for low-order Taylor series approximations of the Xi function Ξ ( z ) as equivalent to the Riemann zeta function ζ ( z ) and the same of the modified
Summary
The present paper tries to find out the common ground for the zeros of the Riemann zeta function ζ = ( z) ζ ( x + iy) and of the modified Bessel functions. Iν ( z) (or Bessel functions Jν ( z ) of imaginary argument z) for imaginary argument z and, for the absence of zeros of the Gaussian Bell ( ) function exp z2. For the function called Riemann zeta function ζ ( z ). Which was known already to Euler but was extended by Riemann to the complex plane Riemann expressed the hypothesis that all nontrivial zeros of this function lie on the axis z=.
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