Siegel–Walfisz theorem

In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz[1] as an application of a theorem by Carl Ludwig Siegel[2] to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.

Statement

Define

${\displaystyle \psi (x;q,a)=\sum _{n\,\leq \,x \atop n\,\equiv \,a\!{\pmod {\!q}}}\Lambda (n),}$

where ${\displaystyle \Lambda }$ denotes the von Mangoldt function, and let φ denote 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

${\displaystyle \psi (x;q,a)={\frac {x}{\varphi (q)}}+O\left(x\exp \left(-C_{N}(\log x)^{\frac {1}{2}}\right)\right),}$

whenever (a, q) = 1 and

${\displaystyle q\leq (\log x)^{N}.}$

Remarks

The constant C_N is not effectively computable because Siegel's theorem is ineffective.

From the theorem we can deduce the following bound regarding the prime number theorem for arithmetic progressions: If, for (a, q) = 1, by ${\displaystyle \pi (x;q,a)}$ we denote the number of primes less than or equal to x which are congruent to a mod q, then

${\displaystyle \pi (x;q,a)={\frac {{\rm {Li}}(x)}{\varphi (q)}}+O\left(x\exp \left(-{\frac {C_{N}}{2}}(\log x)^{\frac {1}{2}}\right)\right),}$

where N, a, q, C_N and φ are as in the theorem, and Li denotes the logarithmic integral.

References

1. Walfisz, Arnold (1936). "Zur additiven Zahlentheorie. II" [On additive number theory. II]. Mathematische Zeitschrift (in German). 40 (1): 592–607. doi:10.1007/BF01218882. MR 1545584.
2. Siegel, Carl Ludwig (1935). "Über die Classenzahl quadratischer Zahlkörper" [On the class numbers of quadratic fields]. Acta Arithmetica (in German). 1 (1): 83–86.