Siegel-Walfisz theorem

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.

Define

where ${\displaystyle \Lambda }$ 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 C_N depending only on N such that

whenever (a, q) = 1 and

The constant C_N 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

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