Siegel–Walfisz theorem

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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.[1]

Statement[edit]

Define

\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n),

where \Lambda denotes the von Mangoldt function and 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

\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\le(\log x)^N.

Remarks[edit]

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 \pi(x;q,a) we denote the number of primes less than or equal to x which are congruent to a mod q, then

\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 offset logarithmic integral.

References[edit]

  1. ^ Walfisz, Arnold (1936). "Zur additiven Zahlentheorie. II ". Mathematische Zeitschrift 40 (1): 592–607. doi:10.1007/BF01218882.  (German)