Zeta function regularization

Remarks

Here $\lambda_n^{-s} = e^{-s \ln \lambda_n}$, which needs $\lambda_n > 0$. The function $\zeta_\lambda$ is a generalized Dirichlet series.

Example: for $\lambda_n = 2^n$ one has $\zeta_\lambda(s) = \sum_{n \ge 1} 2^{-ns} = \frac{1}{2^s - 1}$ for $\operatorname{Re} s > 0$, and the right side is meromorphic on $\C$. So the regularized sum of $2 + 4 + 8 + \cdots$ is $\frac{1}{2^{-1} - 1} = -2$. For $\lambda_n = n$, $\zeta_\lambda$ is the Riemann zeta function, so the regularized sum of $1 + 2 + 3 + \cdots$ is $\zeta(-1) = -\frac{1}{12}$, and the regularized product $\prod n$ is $e^{-\zeta'(0)} = \sqrt{2\pi}$.

Non-example: the sequence $\lambda_n = (-1)^n n$ fails clause 1, since it has negative terms and $\lambda_n^{-s}$ is not defined by a real logarithm.