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In [[analytic number theory]], the '''Siegel–Walfisz theorem''' was obtained by [[Arnold Walfisz]] as an application of a [[Siegel zero|theorem]] by [[Carl Ludwig Siegel]] to [[primes in arithmetic progression]]s.<ref>{{Cite journal |first=Arnold |last=Walfisz |title={{lang|de|Zur additiven Zahlentheorie. II }} |journal=[[Mathematische Zeitschrift]] |volume=40 |issue=1 |pages=592–607 |year=1936 |doi=10.1007/BF01218882 }} {{de icon}}</ref> | |||
==Statement of the Siegel–Walfisz theorem== | |||
Define | |||
:<math>\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n),</math> | |||
where <math>\Lambda</math> denotes the [[von Mangoldt function]] and ''φ'' to be [[Euler's totient function]]. | |||
Then the theorem states that given any real number ''N'' there exists a positive constant ''C''<sub>''N''</sub> depending only on ''N'' such that | |||
:<math>\psi(x;q,a)=\frac{x}{\varphi(q)}+O\left(x\exp\left(-C_N(\log x)^\frac{1}{2}\right)\right),</math> | |||
whenever (''a'', ''q'') = 1 and | |||
:<math>q\le(\log x)^N.</math> | |||
==Remarks== | |||
The constant ''C''<sub>''N''</sub> is not [[effective results in number theory|effectively computable]] because Siegel's theorem is ineffective. | |||
From the theorem we can deduce the following form of the [[prime number theorem for arithmetic progressions]]: If, for (''a'',''q'')=1, by <math>\pi(x;q,a)</math> we denote the number of primes less than or equal to ''x'' which are congruent to ''a'' mod ''q'', then | |||
:<math>\pi(x;q,a)=\frac{{\rm Li}(x)}{\varphi(q)}+O\left(x\exp\left(-\frac{C_N}{2}(\log x)^\frac{1}{2}\right)\right),</math> | |||
where ''N'', ''a'', ''q'', ''C''<sub>''N''</sub> and φ are as in the theorem, and Li denotes the [[offset logarithmic integral]]. | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Siegel-Walfisz theorem}} | |||
[[Category:Theorems in analytic number theory]] | |||
[[Category:Theorems about prime numbers]] | |||
{{numtheory-stub}} | |||
Revision as of 13:51, 8 June 2013
In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions.[1]
Statement of the Siegel–Walfisz theorem
Define
where denotes the von Mangoldt function and φ to be Euler's totient function.
Then the theorem states that given any real number N there exists a positive constant CN depending only on N such that
whenever (a, q) = 1 and
Remarks
The constant CN is not effectively computable because Siegel's theorem is ineffective.
From the theorem we can deduce the following form of the prime number theorem for arithmetic progressions: If, for (a,q)=1, by we denote the number of primes less than or equal to x which are congruent to a mod q, then
where N, a, q, CN and φ are as in the theorem, and Li denotes the offset logarithmic integral.
References
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