To blather on a bit: In fact, the function g(s) = sum {n=1,oo} of (-1)^(n+1) / n^s is easily seen to be (1 - 2/2^s)zeta(s) for s where (***) is defined: where R(s) > 1. But g(s) converges for Re(s) > 0. Thus zeta(s) may be defined as (*) zeta(s) = g(s)/(1 - 2^(1-s)) for Re(s) > 0. Let xi(s) = zeta(s) gamma(s/2) / pi^(s/2). Then the functional equation for zeta is (**) xi(s) = xi(1-s) where xi is defined, i.e., for s neither 0 nor 1. Since zeta is defined for 0 < Re(s) by (*), the functional equation (**) can be used to continue zeta from Re(s) > 0 to all s (except the pole s = 1). (If zeta were defined only for Re(s) > 1 then (**) would give another definition for Re(s) < 0, but it would not be clear whether these two functions are part of the same whole). Now from (**) we get zeta(-1) = (zeta(2)*gamma(1)/pi)/(gamma(-1/2)/pi^(-1/2)) = ((pi^2 / 6) / pi) / (-2*sqrt(pi)*sqrt(pi)) = -1/12. --Dan --------------------------------------------------- << I've seen it written in several places that since 1+2+3+ . . . = sum{n=1,oo} of 1/n^(-1), Ramanujan thought of this sum, in some sense, as zeta(-1), which turns out (using correct analytic continuation of the Dirichlet series (***) zeta(s) = sum{n=1,oo} of 1/n^s, which converges only for Re(s) > 1) to be -1/12.