Dirichlet's theorem on arithmetic progressions

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

For fixed coprime integers $a,q$, the Prime number theorem in the arithmetic progression gives $\pi (x;q,a)\sim x/(\phi (q)\log x)$. Partial summation expresses the reciprocal-prime sum as $\pi (x;q,a)/x+{\int }_2^x\pi (t;q,a)t^{-2}dt$. The integral of the relative error is $o(\log \log x)$: fix a threshold beyond which that error is at most $\epsilon$, bound its remaining integral by $\epsilon \log \log x$, and then let $\epsilon$ decrease to zero. The modulus is fixed in this statement.

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 $(a,q)=1$ and $q\le {\log }^{A}N$, apply the Siegel–Walfisz theorem to every reduced residue class with error exponent $A+B+2$. The main terms sum to $N{c}_q(a)/\phi (q)$, where the Ramanujan sum satisfies $c_q(a)=\mu (q)$ for coprime $a,q$. Nonunits with nonzero Von Mangoldt function are powers of primes dividing $q$ and contribute at most $\omega (q)\log N$, which is absorbed by the error. For arbitrary $a$, the correct main term is $N{c}_q(a)/\phi (q)$; the reduced numerator hypothesis cannot be omitted.

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.