Infinite Arithmetic Series - Zeta Regularization

Zeta Regularization

The zeta-regularized sum of an arithmetic series of the right form is a value of the associated Hurwitz zeta function,

Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζ_R(0) = −1⁄₂ and 1 + 2 + 3 + 4 + · · · to ζ_R(−1) = −1⁄₁₂, where ζ is the Riemann zeta function, the above form is not in general equal to