Siegel’s theorem on primes in arithmetic progressions

Definition 1

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

π(x;q,a)=pxpa(modq)p prime1

i.e. the number of primes not exceeding x that are congruentMathworldPlanetmath 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


for every 1q(logx)A with gcd(q,a)=1.

Note that it follows from this theorem that the distribution of primes among invertible residue classesMathworldPlanetmath 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


This follows easily from on noting that Li(x)=xlogx+.

Title Siegel’s theorem on primes in arithmetic progressions
Canonical name SiegelsTheoremOnPrimesInArithmeticProgressions
Date of creation 2013-03-22 17:58:29
Last modified on 2013-03-22 17:58:29
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 11N13
Classification msc 11A41
Related topic PrimeNumberTheorem