Analytic Continuation of Bernoulli Numbers, a New Formula for the Riemann Zeta Function, and the Phenonmenon of Scattering of Zeros

1997/05/15 by S. C. Woon, Woon, S. C.
Mathematics · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Differential Equations and Boundary Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #math-ph #math.MP #nlin.CD

paper · pdf · doi:10.48550/arxiv.physics/9705021

16 pages, LaTeX, 11 figures. Animated gifs and associated papers are at http://www.damtp.cam.ac.uk/user/scw21/papers/ . On the mathematical and number theory applications of the method in hep-th/9707206

openalex publication_date 1997/05/15 · arxiv created 1997/07/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The method analytic continuation of operators acting integer n-times to complex s-times (hep-th/9707206) is applied to an operator that generates Bernoulli numbers B_n (Math. Mag. 70(1), 51 (1997)). B_n and Bernoulli polynomials B_n(s) are analytic continued to B(s) and B_s(z). A new formula for the Riemann zeta function zeta(s) in terms of nested series of zeta(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenonmenon of `scatterings' of the zeros of B_s(z) is observed.

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