Prime Number Theorem - Prime Number Theorem For Arithmetic Progressions

Prime Number Theorem For Arithmetic Progressions

Let denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre conjectured, and Vallée-Poussin proved, that, if a and n are coprime, then

\pi_{n,a}(x) \sim \frac{1}{\phi(n)}\mathrm{Li}(x),

where φ(·) is the Euler's totient function. In other words, the primes are distributed evenly among the residue classes modulo n with gcd(a, n) = 1. This can be proved using similar methods used by Newman for his proof of the prime number theorem.

The Siegel–Walfisz theorem gives a good estimate for the distribution of primes in residue classes.

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