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.

Statement

Define

$\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n),$

where $\Lambda$ 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_N depending only on N such that

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

whenever (a, q) = 1 and

$q\le(\log x)^N.$

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

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

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