Siegel’s theorem on primes in arithmetic progressions

Definition 1

For $x\mathrm{>}\mathrm{0}$ real, $q\mathrm{,}a$ positive integers, we define

$$\pi (x;q,a)=\sum _{\begin{array}{c}p\le x\\ p\equiv a(modq)\\ p\mathit{\text{ prime}}\end{array}}1$$

i.e. the number of primes not exceeding $x$ that are congruent to $a$ modulo $q$.

Then the following holds:

Theorem 1

(Siegel) For all $A\mathrm{>}\mathrm{0}$, there is some constant $c\mathrm{=}c\mathit{}\mathrm{(}A\mathrm{)}\mathrm{>}\mathrm{0}$ such that

$$\pi (x;q,a)=\frac{Li(x)}{\phi (q)}+\mathrm{O}\left(x\exp (-c\sqrt{\log x})\right)$$

for every $\mathrm{1}\mathrm{\le }q\mathrm{\le }{\mathrm{(}\log \mathit{}x\mathrm{)}}^A$ with $\gcd \mathit{}\mathrm{(}q\mathrm{,}a\mathrm{)}\mathrm{=}\mathrm{1}$.

Note that it follows from this theorem that the distribution of primes among invertible residue classes $modq$ does not depend on the residue class - that is, primes are evenly distributed into such classes.

A form of Dirichlet’s theorem on primes in arithmetic progressions states that

$$\frac{\pi (x;q,a)}{\pi (x)}\sim \frac{1}{\phi (q)}$$

This follows easily from on noting that $Li(x)=\frac{x}{\log x}+\mathrm{\cdots }$.