Proof of Riemann's zeta-hypothesis

Details Proof of Riemann's zeta-hypothesis Proof of Riemann's zeta-hypothesis

2008-09-30

arxiv.org/abs/0809.5120v15

Mathematics

General Mathematics

Proof (21 pages) plus Frequently Asked Questions (57 pages); Appendix C added, FAQ #7a revised

Scientific paper

Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation. Using residue calculus, we can in this way get two alternative, equivalent series expansions for zeta(s) of order N, both valid inside the "critical strip", i e for 0 < Re(s) < 1. Together, these two expansions embody important characteristics of the zeta-function in this range, and their detailed behavior as N tends to infinity can be used to prove Riemann's zeta-hypothesis that the nontrivial zeros of the zeta-function must all have real part 1/2. In addition to the preprint, the arXiv file also contains a discussion of some forty Frequently Asked Questions from readers. Further questions not adequately dealt with in the existing FAQ are welcome.

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