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.
Contents
Statement
Define
where
Then the theorem states that given any real number N there exists a positive constant CN depending only on N such that
whenever (a, q) = 1 and
Remarks
The constant CN 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
where N, a, q, CN and φ are as in the theorem, and Li denotes the offset logarithmic integral.