What is Dirichlet’s theorem about primes in arithmetic progression? What can you say about the density of such primes?
Solution
Let \(q\ge2\) and let \(a\) be an integer with \(\gcd(a,q)=1\).
1. Dirichlet’s theorem says that there are infinitely many primes \(p\) satisfying\[ p\equiv a\pmod q. \]
This is Dirichlet’s theorem on primes in arithmetic progressions: every reduced residue class modulo \(q\) contains infinitely many primes. The coprimality condition is necessary, since if \(\gcd(a,q)>1\), then all sufficiently large integers in the progression share a nontrivial divisor with \(q\).
2. A stronger asymptotic theorem gives\[ \pi(x;q,a)\sim \frac{\operatorname{Li}(x)}{\varphi(q)}, \]where \(\pi(x;q,a)\) counts primes \(p\le x\) with \(p\equiv a\pmod q\).
This is the prime-number theorem for arithmetic progressions. It refines Dirichlet’s infinitude theorem by giving the asymptotic frequency of primes in each reduced residue class.
3. Consequently the primes are asymptotically equidistributed among the \(\varphi(q)\) reduced residue classes modulo \(q\).
The ordinary prime-number theorem gives \[ \pi(x)\sim\operatorname{Li}(x). \] Therefore 2 implies \[ \frac{\pi(x;q,a)}{\pi(x)}\longrightarrow\frac1{\varphi(q)}. \] Thus the relative natural density among all primes is \(1/\varphi(q)\). The same value is also the Dirichlet density.