2008/09/30 by Arne Bergström, Bergstrom, Arne
Medicine · Physics and Astronomy · #11M26 #30E20 #Advanced Thermodynamics and Statistical Mechanics #Biofield Effects and Biophysics #FOS: Mathematics #General Mathematics (math.GM) #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.0809.5120
openalex publication_date 2008/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.