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 in arithmetic progressions.

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" for Euler'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