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On the Essence of the Riemann Zeta Function and Riemann Hypothesis
Version 1
: Received: 24 September 2024 / Approved: 25 September 2024 / Online: 26 September 2024 (03:58:01 CEST)
How to cite: Wang, S. On the Essence of the Riemann Zeta Function and Riemann Hypothesis. Preprints 2024, 2024092040. https://doi.org/10.20944/preprints202409.2040.v1 Wang, S. On the Essence of the Riemann Zeta Function and Riemann Hypothesis. Preprints 2024, 2024092040. https://doi.org/10.20944/preprints202409.2040.v1
Abstract
Riemann’s functional equation is valid on the vertical line of s=1/2. Each side is a real-valued function. The Riemann’s Xi function is also a real-valued function along the vertical line of s=1/2. Through the holomorphic extensions of the Riemann zeta function, starting from the real-valued function at s=1/2 into both sides of sigma<1/2 and sigma>1/2, we can get two versions of the zeta functional equation, eq. (45). The key property of the scaling and rotational factors g(s) and g(1-s) behave as multiplicative inverses in the complex plane, eq. (48). It is deduced that the Zeta function also has multiplicative inverses, the symmetric point is at (1/2,0) in the complex plane. The moduli behave as a hyperbola. Especially, along the vertical line sigam=1/2, the amplitudes of both function g(s) and g(1-s) are equal to 1, its arguments have opposite signs. If sigma is not equal to 1/2, the amplitudes of zeta(s) and zeta(1-s) are not equal to each other, because of their multiplicative inversion relationship. It is deduced that the non-trivial zeros can only be on the vertical line of s=1/2. A gamma function vector field is given in Appendix B, and some moduli of gamma function at special points are given. Finally, another variation of the Zeta function is provided in an integral form in Appendix D. The asymptotes behave as a c8 cyclic group for the large t values.
Keywords
Riemann Zeta function; Riemann hypothesis; Fourier series; Gamma function; multiplicative inversion relationship in the complex plane. Euler product formula for prime numbers
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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