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
where denotes the von Mangoldt function and define φ to be Euler's totient function.
Then the theorem states that given any real number N there exists a positive constant CN depending only on N such that
whenever = 1 andRemarks
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 =1, by we denote the number of primes less than or equal to x which are congruent to a mod q, then
where N, a, q, CN and φ are as in the theorem, and Li denotes the logarithmic integral.