Linnik's theorem is a result in number theory that pertains to the distribution of prime numbers in arithmetic progressions. Specifically, it concerns the distribution of primes in progressions of the form \( a \mod q \) where \( a \) and \( q \) are coprime integers.
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There is an absolute finite exponent bounding the least prime in any reduced arithmetic progression by a constant times . The multiplicative large sieve inequality is one of the tools in proofs of this theorem.