Siegel–Walfisz theorem

The Siegel–Walfisz theorem gives a uniform prime number theorem for arithmetic progressions when the modulus is bounded by a fixed power of the logarithm of the range.

Prime number theorem for arithmetic progressions: A theorem describing the asymptotic distribution of primes in residue classes coprime to a modulus. The Siegel–Walfisz theorem is its uniform version for moduli growing only polylogarithmically.
Prime number theorem: A theorem stating that the number of primes up to x is asymptotic to x divided by log x. The theorem refines this global estimate by separating primes into arithmetic progressions.
Bombieri–Vinogradov theorem: A theorem giving an averaged prime number theorem for arithmetic progressions over moduli up to nearly the square root of the range. It reaches much larger moduli on average, while Siegel–Walfisz gives uniform control for each small modulus.
Generalized Riemann hypothesis: The conjecture that nontrivial zeros of Dirichlet L-functions lie on the line with real part one half. Assuming it gives effective progression estimates over wider modulus ranges than this unconditional theorem.
Dirichlet character: A periodic, completely multiplicative function used to study residue classes modulo an integer. Character sums convert prime counts in individual residue classes into analytic estimates.
Dirichlet's theorem on arithmetic progressions: A theorem stating that every residue class coprime to a modulus contains infinitely many primes. It supplies the qualitative existence result that the Siegel–Walfisz theorem sharpens quantitatively.
Brun–Titchmarsh theorem: An upper-bound sieve estimate for primes in arithmetic progressions, valid across a broader range of moduli. It offers effective upper bounds where the Siegel–Walfisz asymptotic range is too narrow.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one half. It is the principal-character counterpart to the L-function zero questions behind progression estimates.
Dirichlet L-function: A complex function formed from a Dirichlet character and a Dirichlet series, with analytic continuation and a functional equation. The theorem’s proof relies on zero-free regions for these functions.
Euler's totient function: The function φ(q) counts residue classes modulo q that are coprime to q. The expected prime count in each eligible class is normalized by φ(q).

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