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

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

1997-05-15

arxiv.org/abs/physics/9705021v2

Physics

Mathematical Physics

16 pages, LaTeX, 11 figures. Animated gifs and associated papers are at http://www.damtp.cam.ac.uk/user/scw21/papers/ . On the

Scientific paper

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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