Wiener–Ikehara theorem

《Algebraic Number Theory》 · 第 260 页 · Section 8.3「The Wiener–Ikehara Tauberian theorem」, Theorem 8.6

Theorem 8.6 (Wiener–Ikehara) Suppose that the function $a(u)$ is non-negative and increasing on $[0,\infty)$, that $\alpha(s)=\int_0^\infty e^{-us}\,da(u)$ converges for $\sigma>1$, and that $\alpha(s)-c/(s-1)$ extends continuously to $\sigma\ge1$; then $\int_0^x da(u)=ce^x+o(e^x)$.