# Siegel’s theorem on primes in arithmetic progressions

###### Definition 1

For $x>0$ real, $q,a$ positive integers, we define

 $\pi(x;q,a)=\sum_{\begin{subarray}{c}p\leq x\\ p\equiv a\pmod{q}\\ p\text{ prime}\end{subarray}}1$

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 $A>0$, there is some constant $c=c(A)>0$ such that

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

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

Note that it follows from this theorem that the distribution of primes among invertible residue classes $\mod q$ 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

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

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

Title Siegel’s theorem on primes in arithmetic progressions SiegelsTheoremOnPrimesInArithmeticProgressions 2013-03-22 17:58:29 2013-03-22 17:58:29 rm50 (10146) rm50 (10146) 4 rm50 (10146) Theorem msc 11N13 msc 11A41 PrimeNumberTheorem