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
ψ ( x ; q , a ) = ∑ n ≤ x n ≡ a ( mod q ) Λ ( n ) , {\displaystyle \psi (x;q,a)=\sum _{n\,\leq \,x \atop n\,\equiv \,a\!{\pmod {\!q}}}\Lambda (n),}
where Λ {\displaystyle \Lambda } denotes the von Mangoldt function, and let φ {\displaystyle \varphi } 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
ψ ( x ; q , a ) = x φ ( q ) + O ( x exp ( − C N ( log x ) 1 2 ) ) , {\displaystyle \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 ≤ ( log x ) N . {\displaystyle 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)} we denote the number of primes less than or equal to x which are congruent to a mod q, then
π ( x ; q , a ) = L i ( x ) φ ( q ) + O ( x exp ( − C N 2 ( log x ) 1 2 ) ) , {\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.
See also Bombieri–Vinogradov theorem
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