Wiener-Ikehara Tauberian Theorem

The Wiener-Ikehara theorem is a powerful Tauberian theorem that deduces asymptotic behavior of sums from analytic properties of their Dirichlet series generating functions.

DefinitionTauberian Theorems

A Tauberian theorem deduces properties of a sequence from properties of its transform (generating function, Dirichlet series, etc.), typically under additional "Tauberian conditions."

The name comes from Alfred Tauber, who studied conditions under which Abel summability implies Cesàro summability.

TheoremWiener-Ikehara Theorem

Let $f(x) \geq 0$ be non-decreasing, and let:

F(s) = \int_1^{\infty} f(x) x^{-s-1} dx

converge for $\Re(s) > 1$ . Suppose $F(s)$ extends continuously to the closed half-plane $\Re(s) \geq 1$ except for a simple pole at $s=1$ with residue $A$ .

Then:

f(x) \sim Ax \quad \text{as } x \to \infty

ExampleApplying to Prime Number Theorem

For $f(x) = \psi(x) = \sum_{n \leq x} \Lambda(n)$ :

F(s) = \int_1^{\infty} \psi(x) x^{-s-1} dx = -\frac{\zeta'(s)}{\zeta(s)} - \frac{1}{s-1}

The key facts:

$\zeta(s)$ has a simple pole at $s=1$ with residue $1$
$\zeta(s) \neq 0$ for $\Re(s) = 1$ (zero-free region)
Thus $-\zeta'(s)/\zeta(s)$ extends continuously to $\Re(s) \geq 1$ with a simple pole at $s=1$ with residue $1$

Wiener-Ikehara immediately gives $\psi(x) \sim x$ , which is equivalent to PNT.

Remark

The power of the Wiener-Ikehara theorem is that it reduces the Prime Number Theorem to:

Establishing the zero-free region $\zeta(1+it) \neq 0$
Verifying continuity/growth estimates on $\Re(s) = 1$

This is a much more tractable analytic problem than direct estimation of $\psi(x)$ .

DefinitionMel lin Transform Perspective

The Wiener-Ikehara theorem is a Tauberian theorem for Mellin transforms. The Mellin transform:

\mathcal{M}[g](s) = \int_0^{\infty} g(x) x^{s-1} dx

relates function behavior at $x \to \infty$ to transform behavior near the boundary of convergence.

The Wiener-Ikehara theorem exemplifies how analytic continuation and singularity analysis translate into asymptotic information about arithmetic functions—a central theme in analytic number theory.