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

摘要

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