20
Riemann Zeta Function
Abstract
Topic of this chapter is the Riemann zeta function and its non-trivial zeros. The
evaluation of the Riemann zeta function is based on a series expansion and if
necessary additionally on transforming the function argument. The first 50 nontrivial zeros are table based, additional non-trivial zeros will be numerically
evaluated. Corresponding code for evaluating the Riemann zeta function and
computing the non-trivial zeros can be downloaded.
Topic of this chapter is the Riemann zeta function [1,3] and its non-trivial zeros. The
Riemann zeta function plays an important rôle in number theory especially related to
prime numbers. In theoretical physics the zeta function is applied to regularizations.
Applications can be found in quantum dynamics, e.g., the Casimir effect, spectral
distributions of quantum chaotic systems [4], or in string theory [6].
Function Overview
The SPECFUNPHYS class riemzeta returns the function values of the Riemann
zeta function. The SPECFUNPHYS class riemzetaroot supports the computations of the so-called non-trivial zeros, related to the function argument s =
1
2 + it
of the zeta function. 1 riemzetaroot comes in addition with a method to test the
computed zeros. In addition, in the corresponding program folder lives a data file
riemnulllist.mat with the first 50 non-trivial zeros [5].
1 The Riemann hypothesis asserts that all non-trivial zeros lie in the line with real part of s equal
0.5.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_20
233
Riemann Zeta Function
Abstract
Topic of this chapter is the Riemann zeta function and its non-trivial zeros. The
evaluation of the Riemann zeta function is based on a series expansion and if
necessary additionally on transforming the function argument. The first 50 nontrivial zeros are table based, additional non-trivial zeros will be numerically
evaluated. Corresponding code for evaluating the Riemann zeta function and
computing the non-trivial zeros can be downloaded.
Topic of this chapter is the Riemann zeta function [1,3] and its non-trivial zeros. The
Riemann zeta function plays an important rôle in number theory especially related to
prime numbers. In theoretical physics the zeta function is applied to regularizations.
Applications can be found in quantum dynamics, e.g., the Casimir effect, spectral
distributions of quantum chaotic systems [4], or in string theory [6].
Function Overview
The SPECFUNPHYS class riemzeta returns the function values of the Riemann
zeta function. The SPECFUNPHYS class riemzetaroot supports the computations of the so-called non-trivial zeros, related to the function argument s =
1
2 + it
of the zeta function. 1 riemzetaroot comes in addition with a method to test the
computed zeros. In addition, in the corresponding program folder lives a data file
riemnulllist.mat with the first 50 non-trivial zeros [5].
1 The Riemann hypothesis asserts that all non-trivial zeros lie in the line with real part of s equal
0.5.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_20
233