Prime number theorem for arithmetic progressions

For fixed modulus q and residue class a coprime to q, primes congruent to a modulo q have asymptotic density 1/φ(q) among the primes, where φ is Euler’s totient function.

Dirichlet character: A periodic, completely multiplicative arithmetic function used to encode congruence classes modulo an integer. Characters turn prime counts in individual residue classes into sums suited to analytic methods.
Congruence relation: An equivalence relation on integers in which two numbers are equivalent when their difference is divisible by a fixed modulus. The theorem groups primes according to congruence modulo q.
Peter Gustav Lejeune Dirichlet: A German mathematician whose work established foundational results in number theory and mathematical analysis. His theorem on primes in arithmetic progressions introduced characters and L-series into the subject.
Equidistribution: The property that a sequence or set is distributed uniformly across specified classes or regions in an asymptotic sense. For fixed q, primes become equidistributed among the φ(q) reduced residue classes.
Linnik's constant: The least exponent known to suffice in a uniform power bound for the smallest prime in a reduced residue class. The theorem’s fixed-modulus scope leaves the sharp uniform exponent in Linnik’s bound unresolved.
Dirichlet L-function: A complex function formed by a Dirichlet character’s weighted Dirichlet series, with analytic continuation and an Euler product. The theorem’s error term and proof depend on zeros of these functions.
Arithmetic progression: A sequence of numbers formed by repeatedly adding a fixed common difference. The prime classes have the form a, a+q, a+2q, and so on.
Bernhard Riemann: A German mathematician whose work on the zeta function shaped analytic number theory. His ideas about analytic continuation and zeros informed later prime-distribution methods.
Chebyshev bias: The observed tendency for certain residue classes to contain more primes than competing classes over many ranges. Asymptotic equality does not prevent substantial finite-range imbalances between classes.
Siegel zero: A hypothetical real zero of a Dirichlet L-function unusually close to 1. Such zeros obstruct effective uniform error estimates across varying moduli.

Linked from 10 pages

Brun–Titchmarsh theoremCompared with: Its asymptotic precision exceeds Brun–Titchmarsh when the modulus is fixed and x grows.