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
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 CN depending only on N such that
whenever (a,q)=1 and
[edit] Remarks
The constant CN 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
where N, a, q, CN and φ are as in the theorem, and Li denotes the offset logarithmic integral.
[edit] References
- ^ Mathematische Zeitschrift, 40, pages 592-607, 1936
