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An Elementary Proof of the Prime-Number Theorem for Arithmetic Progressions

Published online by Cambridge University Press:  20 November 2018

Atle Selberg
Atle Selberg*
Affiliation:
The Institute for Advanced Study and Syracuse University
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In this paper we shall give an elementary proof of the theorem

(1.1)

where φ(k) denotes Euler's function, and

(1.2)

where p denotes the prime, and and are integers with (,) = 1, positive.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 66 - 78
Copyright
Copyright © Canadian Mathematical Society 1950

References

1 “An Elementary Proof of the Prime-number Theorem,” Ann. of Math., vol. 50 (1949), 305–313.Google Scholar

2 “An Elementary Proof of Dirichlet's Theorem about Primes in an Arithmetic Progression,” Ann. of Math., vol. 50 (1949), 297–304.Google Scholar

3 We write instead of where no misunderstanding can occur.Google Scholar

4 Instead of (2.8) we might use the somewhat sharper inequality

which can be proved in a similar way.

5 See for instance Dirichlet-Dedekind: Vorlesungen iiber Zahlentheorie (the beginning of §135)

6 For example, by showing that the number of terms with .

7 For example, by noting that a “period-parallelogram“ may always be chosen so that neither of its sides is greater than a diagonal.

8 Or, otherwise expressed, that the lattice may be built up of “period-parallelograms“ with both sides .

9 By residues, we understand here residues belonging to the reduced residue system.

10 By values we mean here residues mod .

11 For there is then a y in the interval with .

12 For there is then a y in the interval with .