Siegel–Walfisz theorem

Mathematician Arnold Walfisz, 1920 at Göttingen.

In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel 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

$\psi (x;q,a)=\sum _{\genfrac{}{}{0ex}{}{n\le x}{n\equiv a(\mathrm{mod}q)}}\mathrm{\Lambda }(n),$

where $\mathrm{\Lambda }$ denotes the von Mangoldt function, and let $\phi$ 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

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

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

$\pi (x;q,a)=\frac{\mathrm{L}\mathrm{i}(x)}{\phi (q)}+O(x\exp (-\frac{C_N}{2}(\log x{)}^{\frac{1}{2}})),$

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

See also

• Bombieri–Vinogradov theorem