Siegel–Walfisz Theorem

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

:$\psi(x;q,a) = \sum_\Lambda(n),$

where $\Lambda$ 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

:$\psi(x;q,a)=\frac+O\left(x\exp\left(-C_N(\log x)^\frac\right)\right),$

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
:$\pi(x;q,a) = \frac+O\left(x\exp\left(-\frac(\log x)^\frac\right)\right),$
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

{{DEFAULTSORT:Siegel-Walfisz theorem
Theorems in analytic number theory
Theorems about prime numbers