---
tag: 63FS7T8R18
kind: definition
name: Zeta function regularization
msc: ["11M41", "58J52", "40G99"]
defines: VJCJYXGTDY
uses: [0QE8STGJH2, 8MWTEF6462, 0M3YQJ81NB]
sources:
  - url: https://en.wikipedia.org/wiki/Zeta_function_regularization
---

## Definition

Let $(\lambda_n)_{n \ge 1}$ be a [[sequence|0QE8STGJH2]] of real numbers such that:

1. $0 < \lambda_1 \le \lambda_2 \le \cdots$ and $\lambda_n \to \infty$;
2. the series $\zeta_\lambda(s) = \sum_{n \ge 1} \lambda_n^{-s}$ converges for all complex $s$ with $\operatorname{Re} s > \sigma_0$, for some real $\sigma_0$;
3. $\zeta_\lambda$ has a meromorphic [[analytic continuation|0M3YQJ81NB]] to a domain that contains $s = -1$ and $s = 0$.

If the continuation is holomorphic at $s = -1$, the *zeta-regularized sum* of $\sum \lambda_n$ is $\zeta_\lambda(-1)$.
If it is holomorphic at $s = 0$, the *zeta-regularized product* of $\prod \lambda_n$ is $\exp(-\zeta_\lambda'(0))$.

## Remarks

Here $\lambda_n^{-s} = e^{-s \ln \lambda_n}$, which needs $\lambda_n > 0$.
The function $\zeta_\lambda$ is a generalized [[Dirichlet series|8MWTEF6462]].

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.