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.
Statement
Define
\( {\displaystyle \psi (x;q,a)=\sum _{n\,\leq \,x \atop n\,\equiv \,a\!{\pmod {\!q}}}\Lambda (n),} \)
where \( \Lambda \) denotes the von Mangoldt function, and let φ denote 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
\( \psi (x;q,a)={\frac {x}{\varphi (q)}}+O\left(x\exp \left(-C_{N}(\log x)^{{\frac {1}{2}}}\right)\right), \)
whenever (a, q) = 1 and
\( q\leq (\log x)^{N}. \)
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 π ( x ; q , a ) {\displaystyle \pi (x;q,a)} \pi (x;q,a) we denote the number of primes less than or equal to x which are congruent to a mod q, then
\( {\displaystyle \pi (x;q,a)={\frac {{\rm {Li}}(x)}{\varphi (q)}}+O\left(x\exp \left(-{\frac {C_{N}}{2}}(\log x)^{\frac {1}{2}}\right)\right),} \)
where N, a, q, CN and φ are as in the theorem, and Li denotes the logarithmic integral.
References
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.
Siegel, Carl Ludwig (1935). "Über die Classenzahl quadratischer Zahlkörper" [On the class numbers of quadratic fields]. Acta Arithmetica (in German). 1 (1): 83–86.
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