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.

Statement of the Siegel–Walfisz theorem

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 is 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

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

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

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

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