Dirichlet's Theorem

Theorem

For any positive integers $a$ and $m$ such that $(a,m)=1$ , there exists infinitely many prime $p$ such that $p\equiv a\mod m$

Hence, for any arithmetic progression, unless it obviously contains finitely many primes (first term and common difference not coprime), it contains infinitely many primes.

Stronger Result

For any positive integers $a$ and $m$ such that $(a,m)=1$ , $\sum_{\substack{p\leq x\\ p\equiv a\mod m}}\frac{1}{p}=\frac{1}{\phi(m)}\log\log x+O(1)$ where the sum is over all primes $p$ less than $x$ that are congruent to $a$ mod $m$ , and $\phi(x)$ is the totient function.

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Dirichlet's Theorem

Theorem

Stronger Result

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