The Siegel-Walfisz Theorem refines the prime number theorem for arithmetic progressions, giving an asymptotic count of primes up to a bound within a fixed residue class modulo an integer, with an error term that holds uniformly as the modulus grows slowly relative to the bound. Named for Carl Ludwig Siegel and Arnold Walfisz, it strengthens Dirichlet's theorem on primes in arithmetic progressions from a statement of infinitude to a precise asymptotic count, though its constants are famously non-effective.
Facts
StatementA refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions, obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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Source Siegel-Walfisz theorem, Wikipedia
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Source Siegel-Walfisz theorem, Wikipedia
Sources
1. Siegel-Walfisz theorem, Wikipedia
Lead paragraph
It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.
- In Branch: Analytic Number Theory, Lead sentence
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