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The prime number theorem via the large sieve

Published online by Cambridge University Press:  26 February 2010

Adolf Hildebrand
Adolf Hildebrand
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801, U.S.A.
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Extract

In the last three decades there appeared a number of elementary proofs of the prime number theorem (PNT) in the literature (see [3] for a survey). Most of these proofs are based, at least in part, on ideas from the original proof by Erdős [5] and Selberg [12]. In particular, one of the main ingredients of the Erdős-Selberg proof, Selberg's formula

(where p and q run through primes) appears, in some form, in almost all these proofs.

Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 23 - 30
Copyright
Copyright © University College London 1986

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References

1.Corrádi, K.. A remark on the theory of multiplicative functions. Acta Sci. Math. (Szeged), 28 (1967), 83–92.Google Scholar
2.Daboussi, H.. Sur le Théorème des Nombres Premiers. C. R. Acad. Sc. Paris, Série I, 298 (1984), 161–164.Google Scholar
3.Diamond, H.. Elementary methods in the study of the distribution of prime numbers. Bull Amer. Math. Soc., 7 (1982), 553–589.CrossRefGoogle Scholar
4.Elliott, P. D. T. A.. Probabilistic Number Theory I (Springer, New York, 1979).CrossRefGoogle Scholar
5.Erdős, P.. On a new method, which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 374–384.CrossRefGoogle Scholar
6.Halberstam, H. and Richert, H.-E.. Sieve Methods (Academic Press, London, 1974).Google Scholar
7.Hildebrand, A.. On Wirsing's mean value theorem for multiplicative functions. Bull. London Math. Soc., 18 (1986), 147–152.CrossRefGoogle Scholar
8.Kalecki, M.. A simple elementary proof of M(x) = ∑n≤x μ(n) = o(x). Acta Arith., 13 (1967), 1–7.CrossRefGoogle Scholar
9.Montgomery, H. L.. Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, Vol. 227 (Springer, Berlin, 1971).CrossRefGoogle Scholar
10.Postnikov, A. G. and Romanov, N. P.. A simplilcation of Selberg's elementary proof of the asymptotic law of distribution of primes. Uspehi Mat. Nauk, 10 (1955), 75–87.Google Scholar
11.Renyi, A.. On the large sieve of Ju. V. Linnik. Compositio Math., 8 (1950), 68–75.Google Scholar
12.Selberg, A.. An elementary proof of the prime number theorem. Ann. Math., 50 (1949), 305–313.CrossRefGoogle Scholar