ZETA-FUNCTION AND ETA-FUNCTION RESUMMATION OF INFINITE SERIES - GENERAL-CASE

Given a spectrum of positive numbers {lambda-m} from which a zeta-function Z(s) = SIGMA-m-lambda-m(-s) can be constructed, the reorganization of series of the type [GRAPHICS] into power series in t is examined in detail using the method of zeta-function resummation. For summand functions f(lambda-m-t) having power series expansions in lambda-m-t with infinite radius of convergence, and which satisfy other conditions of a rather general nature, we find that F(s, t) can be reorganized to [GRAPHICS] where R(s, t) vanishes exponentially as t --> 0. The numbers a(n), b(n), c(n), d(n) can all be computed in terms of the zeta-function Z(s). R(s, t) is difficult to evaluate, but important general features of this function can be determined. The power series expansion of F(s, t) can be regarded as a generalization of the heat kernel expansion (for which F(lambda-m-t) = exp)-lambda-m-t) and s = 0) to non-zero complex variable s (which is useful) and to many other summand functions f(lambda-m-t). Remarkably, the zeta-function resummation method can be applied as easily to divergent series F(s, t) as it can to convergent ones. The method is therefore both a rearrangement procedure for convergent series, and a summation prescription for divergent series.