Analytic Continuation of Bernoulli Numbers, a New Formula for the Riemann Zeta Function, and the Phenonmenon of Scattering of Zeros
S. C. Woon
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 Bn (Math. Mag. 70(1), 51 (1997)). Bn and Bernoulli polynomials Bn(s) are analytic continued to B(s) and Bs(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 Bs(z) is observed.
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