 Siegel–Walfisz theorem

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.

The Siegel–Walfisz theorem is a result in analytic number theory that provides a relationship between the distribution of prime numbers and certain arithmetic functions. Specifically, it deals with the distribution of prime numbers in arithmetic progressions and offers an asymptotic formula for the count of such primes.