Harman Patil (Editor)

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.

Contents

Statement

Define

ψ ( x ; q , a ) = ∑ n ≤ x n ≡ a ( mod q ) Λ ( n ) ,

where Λ 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

ψ ( x ; q , a ) = x φ ( q ) + O ( x exp ⁡ ( − C N ( log ⁡ x ) 1 2 ) ) ,

whenever (a, q) = 1 and

q ≤ ( 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 ) 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 ) ) ,

where N, a, q, CN and φ are as in the theorem, and Li denotes the offset logarithmic integral.

References

Siegel–Walfisz theorem Wikipedia
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