Multiplicities for Riemann ξ Function and its Derivatives in the Critical Strip and Analysis for Their Nontrivial Zeros

Liu Xiang; Liu Fasheng

doi:https://doi.org/10.14445/22315373/IJMTT-V66I5P521

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 5 | Year 2020 | Article Id. IJMTT-V66I5P521 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I5P521

Multiplicities for Riemann ξ Function and its Derivatives in the Critical Strip and Analysis for Their Nontrivial Zeros

Liu Xiang, Liu Fasheng

Abstract

The multiplicity for Riemann ξ function in the critical strip 0 < σ < 1 is presented based on the intermediate value theorem, and, similarly, the result is also true for Riemann ξ function's derivatives, and then the nontrivial zeros of Riemann ξ function are analyzed. All the nontrivial zeros fall on the critical line with the real part of 1/2, So the Riemann hypothesis is proven true and all the propositions equivalent to Riemann hypothesis and the conclusions based on Riemann hypothesis are all true.

Keywords

Riemann ξ function, Euler production, analytical continuation, second mean-value theorem, nontrivial zeros, Riemann hypothesis (RH).

References

[1] Riemann B.,Uber die Anzahl der Primzahlen unter einer gegebenen Grosse. Monats. Preuss. Akaad, Wiss.(1859-1860),671-680.
[2] Peter Browein, Stephen Choi, Brendan Rooney and Andrea Wei rathmueller Riemann Hypothesis, A Resource for the Afficionado and Virtuoso Alike. Springer 2008.
[3] A.A.Karatsuba, S.M. Voronin Translated from the Russian by Neal Koblitz, The Riemann Zeta-Function. Walter de Gruyter . Berlin. New York. 1992.

Citation :

Liu Xiang, Liu Fasheng, "Multiplicities for Riemann ξ Function and its Derivatives in the Critical Strip and Analysis for Their Nontrivial Zeros," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 5, pp. 155-159, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I5P521