Wiener–Ikehara theorem

Remarks

The proof is Fourier-analytic. One integrates $f(s) - \frac{c}{s-1}$ on vertical lines against a Fejér-type kernel with compactly supported Fourier transform, lets $\operatorname{Re} s \to 1^+$, and uses the Riemann–Lebesgue lemma; monotonicity of $A$ then removes the smoothing.

Example: the prime number theorem. Let $\psi(y) = \sum_{n \le y} \Lambda(n)$ with the von Mangoldt function $\Lambda$, and put $A(x) = \psi(e^x)$. Then $\int_0^\infty \psi(e^x) e^{-xs} \, dx = -\frac{\zeta'(s)}{s \, \zeta(s)}$ for $\operatorname{Re} s > 1$. This differs from $\frac{1}{s - 1}$ by a function continuous on $\operatorname{Re} s \ge 1$, because $\zeta$ has a simple pole at $1$ and does not vanish on the line $\operatorname{Re} s = 1$. So $\psi(y)/y \to 1$, which is equivalent to the prime number theorem.

The hypothesis is only continuity up to the boundary line, not analytic continuation beyond it.

Going deeper

For a student

In Dirichlet series form: if $a_n \ge 0$ and $F(s) = \sum a_n n^{-s}$ converges for $\operatorname{Re} s > 1$ with $F(s) - \frac{c}{s-1}$ continuous on $\operatorname{Re} s \ge 1$, then $\sum_{n \le y} a_n \sim c\,y$. This follows by applying the theorem to $A(x) = \sum_{n \le e^x} a_n$. The theorem is a special case of Wiener's general Tauberian theory; Newman's later contour-integral proof of the prime number theorem uses a closely related Tauberian argument.