Using the
properties of theta-series and Schwarz reflection principle, a proof for
Riemann hypothesis (RH) is directly presented and the first ten nontrivial
zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all
its equivalent results and the consequences assuming RH are true.
References
[1]
Riemann, B. (1859-1860) über die Anzahl der Primzahlen unter einer gegebenen Grosse. Monats. Preuss. Akaad, Wiss., 671-680.
Titchmarsh, E.C. (1986) The Theory of The Riemann Zeta-Function. Clarendon Press, Oxford.
[4]
Keating, J.P. and Snaith, N.C. (2000) Random Matrix Theory And ζ (1/2 + it). Communications in Mathematical Physics, 214, 57-89. http://dx.doi.org/10.1007/s002200000261
[5]
Knauf, A. (1999) Number Theory, Dynamical Systems and Statistical Mechanics. Reviews in Mathematical Physics, 11, 1027.
[6]
Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2008) The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike. Springer, Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo.
[7]
Karatsuba, A.A. and Voronin S.M. (1992) The Riemann Zeta-Function. Translated from the Russian by Neal Koblitz, Walter de Gruyter. Berlin, New York. http://dx.doi.org/10.1515/9783110886146
[8]
Everest, G. (2011) An Introduction to Number Theory. Science Press, Beijing.
