Siegel–Walfisz theorem

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.

Statement Define {displaystyle psi (x;q,a)=sum _{n,leq ,x atop n,equiv ,a!{pmod {!q}}}Lambda (n),} where {displaystyle 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 {displaystyle psi (x;q,a)={frac {x}{varphi (q)}}+Oleft(xexp left(-C_{N}(log x)^{frac {1}{2}}right)right),} whenever (a, q) = 1 and {displaystyle qleq (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 {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 {displaystyle pi (x;q,a)={frac {{rm {Li}}(x)}{varphi (q)}}+Oleft(xexp 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. Categories: Theorems in analytic number theoryTheorems about prime numbers