|
|
| Line 1: |
Line 1: |
| 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>
| | Secondary School Teacher Winfred Deniston from Sainte-Genevieve, enjoys to spend time cycling, [http://ganhedinheiro.comoganhardinheiro101.com como ganhar dinheiro] na internet and handwriting. In recent time took some time to visit Group of Monuments at Mahabalipuram. |
| | |
| ==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}}
| |
Secondary School Teacher Winfred Deniston from Sainte-Genevieve, enjoys to spend time cycling, como ganhar dinheiro na internet and handwriting. In recent time took some time to visit Group of Monuments at Mahabalipuram.