- \(^\ast\) One can combine ideas from the prime number theorem with the proof of Dirichlet’s theorem about primes in arithmetic progression (given in Book I) to prove the following. Let \(q\) and \(\ell\) be relatively prime integers. We consider the primes belonging to the arithmetic progression \(\{ q k + \ell \} _ { k = 1 } ^ { \infty }\) , and let \(\pi _ { q , \ell } ( x )\) denote the number of such primes \(\leq x\) . Then one has
\[ \pi_ {q, \ell} (x) \sim \frac {x}{\varphi (q) \log x} \quad \mathrm{as} x \to \infty , \]