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Über das Primzahl-Zwillingsproblem. (On the twin-prime problem). (English) Zbl 0646.10033
It is shown that for ‘almost all’ numbers \(k\leq x\) and all \(y\in [2x,x^{8/5-\epsilon}]\) the number of prime pairs \(\leq y\) with difference 2k is asymptotically equal to the expression conjectured by Hardy-Littlewood. This improves earlier results of van der Corput and Lavrik.
Reviewer: D.Wolke

11N05 Distribution of primes
Full Text: DOI EuDML
[1] Corput, J.G. van der: Sur l’hypothèse de Goldbach pour presque tous les nombres pairs. Acta Arith.2, 266-290 (1937) · Zbl 0018.05203
[2] Hardy, G.H., Littlewood, J.E.: Some problems of ?Partitio Numerorum?. III: On the expression of a number as a sum of primes. Acta Math.44, 1-70 (1923) · JFM 48.0143.04 · doi:10.1007/BF02403921
[3] Jahnke, Th.: Über die Anzahl der Primzahlpaare mit gegebenem Abstand. Dissertation Univ. Freiburg. 1978
[4] Lavrik, A.F.: On the twin prime hypothesis of the theory of primes by the method of I.M. Vinogradov. Dokl. Akad. Nauk SSSR132, 1013-1015 (1960) = Sov. Math. Dokl. 1, 700-702 (1960) · Zbl 0097.03104
[5] Montgomery, H.L.: Topics in multiplicative number theory. (Lectures Notes Mathematics 227). Berlin-Heidelberg-New York: Springer 1971 · Zbl 0216.03501
[6] Montgomery, H.L., Vaughan, R.C.: The exceptional set in Goldbach’s problem. Acta Arith.27, 353-370 (1975) · Zbl 0301.10043
[7] Prachar, K.: Primzahlverteilung. Berlin Göttingen Heidelberg: Springer 1957
[8] Richert, H.-E.: Lectures on Sieve methods. Bombay: Tata Institute 1976 · Zbl 0392.10041
[9] Titchmarsh, E.C.: The theory of the Riemann zeta-function. Oxford: Clarendon 1951 · Zbl 0042.07901
[10] Vaughan, R.C.: The Hardy-Littlewood method. Cambridge: University Press 1981 · Zbl 0455.10034
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