Siegel–Walfisz theorem

In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz[1] as an application of a theorem by Carl Ludwig Siegel[2] 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.




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 (a, q) = 1 and



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   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.


  1. ^ Walfisz, Arnold (1936). "Zur additiven Zahlentheorie. II" [On additive number theory. II]. Mathematische Zeitschrift (in German). 40 (1): 592–607. doi:10.1007/BF01218882. MR 1545584.
  2. ^ Siegel, Carl Ludwig (1935). "Über die Classenzahl quadratischer Zahlkörper" [On the class numbers of quadratic fields]. Acta Arithmetica (in German). 1 (1): 83–86.