Newman tauberian prime number theorem

Given: The data and hypotheses of the statement.

Dirichlet character chebyshev laplace transform: Fix a Dirichlet character $\chi$ modulo $q\ge 1$. Put ${\mathrm{\Psi }}_{\chi }(x)={\sum }_{n\le x}\chi (n)\mathrm{\Lambda }(n)$ and ${\delta }_{\chi }=1$ for the principal character, zero otherwise. The bounded, locally integrable function $f_{\chi }(t)=e^{-t}{\mathrm{\Psi }}_{\chi }(e^t)-{\delta }_{\chi }$ has Laplace transform $g_{\chi }(s)=-\frac{L^{\mathrm{\prime }}(s+1,\chi )}{(s+1)L(s+1,\chi )}-\frac{{\delta }_{\chi }}{s}(\mathrm{Re}s>0)\mathrm{.}$ After the removable value at zero is filled in, this extends holomorphically to an open neighborhood of the closed right half-plane.

Newman zagier tauberian theorem: Let $f:[0,\mathrm{\infty })\to \mathbb{C}$ be bounded and locally Lebesgue integrable. If $g(z)={\int }_0^{\mathrm{\infty }}f(t)e^{-zt}dt$, initially defined for $\mathrm{Re}z>0$, extends holomorphically to an open set containing $\{\mathrm{Re}z\ge 0\}$, then ${\lim }_{T\to \mathrm{\infty }}{\int }_0^{T}f(t)dt=g(0)\mathrm{.}$

Monotone chebyshev tauberian desmoothing: Let $A:[1,\mathrm{\infty })\to [0,\mathrm{\infty })$ be nondecreasing and locally integrable, with $A(x)=O(x)$, and let $a\ge 0$. If ${\int }_1^{\mathrm{\infty }}(A(x)-ax)x^{-2}dx$ converges, then $A(x)\mathrm{/}x\to a$.