Dirichlet theorem on primes in arithmetic progressions

If $\gcd (a,m)=1$, infinitely many primes satisfy

For fixed $A,C>0$,uniformly for $q\le {\log }^{C}x$ and $(a,q)=1$. The implied constant may be ineffective. This small-modulus input complements the large sieve in the proof of the Bombieri–Vinogradov theorem.

For real $s>1$, the logarithm defined by the Euler product of a Dirichlet L-function differs from $F_{\chi }(s)$ by a quantity of absolute value at most ${\sum }_{n\ge 2}1/(n(n-1))=1$. If $L(\chi ,1)\ne 0$, a local holomorphic logarithm differs from this continuous logarithm by one fixed multiple of $2\pi i$ on a short real interval, so $F_{\chi }(s)$ stays bounded as $s\downarrow 1$. For the principal Dirichlet character, the pole of the Riemann zeta function instead gives $F_{{\chi }_0}(s)=\log (1/(s-1))+O(1)$. Orthogonality of Dirichlet characters then makes the sum over any reduced residue class diverge, proving infinitude of its primes.