Linnik's theorem

Linnik's theorem states that an absolute constant L exists such that every reduced residue class modulo q contains a prime below q^L.

Least prime in an arithmetic progression: The smallest prime congruent to a specified reduced residue class modulo an integer. This is the quantity Linnik's theorem bounds by a fixed power of the modulus.
Arithmetic progression: A sequence of numbers with a fixed difference between consecutive terms. The theorem concerns primes among the terms of a progression with fixed modulus and residue.
Yuri Linnik: A Soviet mathematician known for work in analytic number theory and probability. He proved the first uniform power bound for the least prime in a reduced residue class.
Least quadratic nonresidue: The smallest positive integer that is not a quadratic residue modulo a given odd prime. Character-sum methods related to Linnik's theorem also bound the first integer with prescribed character behavior.
Dirichlet character: A periodic, completely multiplicative function used to encode residue classes modulo an integer. Character sums turn primes in a residue class into analytic objects that can be estimated.
Congruence relation: A relation declaring two integers equivalent when their difference is divisible by a given modulus. It expresses the condition that a prime belongs to a specified residue class.
Linnik's constant: An exponent L for which every reduced residue class modulo q contains a prime below q^L. The theorem's central quantitative feature is the existence of an absolute exponent.
Primes in arithmetic progressions: The study of how primes are distributed among residue classes modulo integers. Linnik's result is a quantitative theorem within this broader subject.
Dirichlet L-function: A complex function formed from a Dirichlet character's series and analytic continuation. Its zeros and values control the prime-counting estimates behind the bound.
Reduced residue system: A set containing one representative of each residue class relatively prime to a modulus. Only classes coprime to the modulus can contain primes beyond its prime divisors.

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