Siegel-Walfisz theorem

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In mathematics, the Siegel-Walfisz theorem was obtained by Arnold Walfisz^[1] as an application of a theorem by Siegel to primes in arithmetic progressions.

[edit] Statement of the Siegel-Walfisz theorem

We define

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

where $Λ$ denotes the von Mangoldt function. We further use the letter φ for 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

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

[edit] 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 $π(x; q, a)$ we denote the number of primes less than or equal to x which are congruent to a mod q, then