Siegel–Walfisz theorem explained

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

\psi(x;q,a) = \sum _{n\leqx} }\Lambda(n),

where

Λ 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

\psi(x;q,a) = (x)/(\varphi(q)) + O\left(x\exp\left( - C _{N(log} x) ^{(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 bound regarding 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) = ({\rm Li)/((x)}{\varphi(q)} + O\left(x\exp\left( - (C _{N})/(2)) ( log x) ^{(1)/(2)} \right)\right), where N, a, q, C_N and φ are as in the theorem, and Li denotes the logarithmic integral.

See also

• Bombieri–Vinogradov theorem

References

1. Arnold . Walfisz . Zur additiven Zahlentheorie. II . On additive number theory. II . . 40 . 1 . 592–607 . 1936 . 10.1007/BF01218882 . de . 1545584 .
2. Siegel . Carl Ludwig . 1935 . . 1 . 1 . 83–86 . Über die Classenzahl quadratischer Zahlkörper . de . On the class numbers of quadratic fields.