Prime number theorem arithmetic progressions

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 ${\lim }_{T\to \mathrm{\infty }}{\int }_0^{T}f(t)dt=g(0)\mathrm{.}$

Orthogonality relations for Dirichlet characters modulo q: Let $G=(\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }$, and let the sum range over all Dirichlet characters modulo $q$. 1. For unit classes $a,b\in G$, ${\sum }_{\chi <mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q}\chi (a)\overline{\chi (b)}=\{\begin{array}{ll}\phi (q),& a=b,\\ 0,& a\mathrm{\ne }b\mathrm{.}\end{array}$ 2. For Dirichlet characters $\chi ,\psi$ modulo $q$, ${\sum }_{a\in G}\chi (a)\overline{\psi (a)}=\{\begin{array}{ll}\phi (q),& \chi =\psi ,\\ 0,& \chi \mathrm{\ne }\psi \mathrm{.}\end{array}$

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 $A(x)\mathrm{/}x\to a$.

Psi and theta differ by at most a square-root term: There are positive constants $K_1,K_2$ such that for every real $x\ge 2$, $0\le \psi (x)-\theta (x)\le K_1\sqrt{x}\log x$ and, for all sufficiently large $x$, $\psi (x)-\theta (x)\le K_2\sqrt{x}\mathrm{.}$

Abel summation recovers the prime-counting function from theta: For every real $x\ge 2$, $\pi (x)=\frac{\theta (x)}{\log x}+{\int }_2^x\frac{\theta (t)}{t{\log }^{2}t}dt\mathrm{.}$

Logarithmic integral asymptotic expansion: For each fixed integer $m\ge 1$, as $x\to \mathrm{\infty }$, $\mathrm{Li}(x)={\sum }_{j=0}^{m-1}\frac{j!x}{{\log }^{j+1}x}+O_m\left(\frac{x}{{\log }^{m+1}x}\right)\mathrm{.}$