- Siegel–Walfisz theorem
In mathematics, the Siegel–Walfisz theorem was obtained by
Arnold Walfisz ["Mathematische Zeitschrift", 40, pages 592-607, 1936] as an application of a theorem by Siegel to primes inarithmetic progression s.tatement of the Siegel–Walfisz theorem
We define
:psi(x;q,a)=sum_{nleq xatop nequiv amod q}Lambda(n),
where Lambda denotes the
von Mangoldt function . We further use the letter "φ" forEuler's totient function .Then the theorem states that given any real number "N" there exists a positive constant "C""N" depending only on "N" such that
:psi(x;q,a)=frac{x}{phi(q)}+Oleft(xexpleft(-C_N(log x)^frac{1}{2} ight) ight),
whenever ("a", "q") = 1 and
:qle(log x)^N.
Remarks
The constant "C""N" 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 pi(x;q,a) we denote the number of primes less than or equal to "x" which are congruent to "a" mod "q", then:pi(x;q,a)=frac m Li}(x)}{phi(q)}+Oleft(xexpleft(-frac{C_N}{2}(log x)^frac{1}{2} ight) ight),where "N", "a", "q", "C""N" and φ are as in the theorem, and Li denotes the
offset logarithmic integral .References
Wikimedia Foundation. 2010.