information:classification
Zeta function regularization

If the series

(for positive values of the a) converges for large real s and can be analytically continued along the real line to s = −1, then its value at s = −1 is called the zeta regularized sum of the series a + a +... Zeta function regularization is nonlinear. In applications, the numbers a are sometimes the eigenvalues of a self-adjoint operator A with compact resolvent, and f (s) is then the trace of A. For example, if A has eigenvalues 1, 2, 3,... then f (s) is the Riemann zeta function, ζ (s), whose value at s = −1 is − 1 / 12, assigning a value to the divergent series 1 + 2 + 3 + 4 +.... Other values of s can also be used to assign values for the divergent sums ζ (0) = 1 + 1 + 1 +... = − 1 / 2, ζ (−2) = 1 + 4 + 9 +... = 0 and in general

where B is a Bernoulli number.