How zeta(s) is eventually defined on (almost) the whole complex plane is interesting. Let s be a complex number. Then for Re(s) > 1 the expression (*) zeta(z) = Sum_{n=1 to oo} 1/n^s converges.* So zeta is now defined for the half-plane Re(s) > 1. Now multiply both sides of (*) by (1 - 2/2^s) times itself to get (**) (1-1/2^(s-1)) zeta(s) = Sum_{n=1 to oo} (-1)^(n+1) / n^s This equation holds for all s with Re(s) > 1, so by permanence it must hold for any s where zeta is defined. Luckily, the RHS of (**) converges for a larger half-plane, namely where Re(s) > 0. So we can solve for zeta as long as we can divide both sides by (1 - 1/2^(s-1)), namely as long as s is unequal to 1. (Which is just as well, since zeta has a pole as s = 1.) zeta(s) = 1/(1-1/2^(s-1)) * Sum_{n=1 to oo} (-1)^(n+1) / n^s for all s with Re(s) > 0 and s unequal to 1. But there is one more trick enabling zeta to be extended to include the left half-plane so zeta(s) is defined for all s unequal to 1. That is the functional equation of zeta in symmetric form, namely, the function xi given by xi(s) = pi^(-s/2) gamma(s/2) zeta(s) satisfies (***) xi(s) = xi(1-s) or written out, pi^(-s/2) gamma(s/2) zeta(s) = pi^((1-s)/2) gamma((1-s)/2) zeta(1-s). A priori this can make sense only on an open set that is invariant under the mapping s —> 1-s, but we have such an open set: the strip {s in C | 0 < Re(s) < 1}. But permanence implies that any functional equation for an analytic function that holds on an open set holds everywhere. So (***) can be used to newly define zeta anywhere that zeta(1-s) = pi^(-s/2) gamma(s/2) zeta(s) / (pi^((1-s)/2) gamma((1-s)/2)) defines a value we hadn't defined before. Since we already know zeta for all s unequal to 1 with Re(s) > 0, this takes care of the remaining s (Re(s) <= 0). —Dan ——————————————————————————————————— * with n^s defined as exp(s * ln(n) Mike Stay wrote: ----- Summing up the naturals gives infinity, but you can analytically continue the Riemann zeta function to -1 and get -1/12. Here's the story about bosonic string theory: http://math.ucr.edu/home/baez/numbers/24.pdf -----