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On the exceptional set for the $2k$-twin primes problem

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On the exceptional set for the 2k-twin primes problem COMPOSITIO MATHEMATICA ALBERTO PERELLI JÁNOS PINTZ On the exceptional set for the 2k-twin primes problem Compositio Mathematica, tome 82, no 3 (1992), p. 355-372. <> © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 355- On the exceptional set for the 2k-twin primes problem ALBERTO PERELLI1 and JANOS PINTZ2* 1 Dipartimento di Matematica, Via L. B. Alberti 4, 16132 Genova, Italy; 2Mathematical Institute of the Hungarian Academy of Sciences, Reàltanoda u. 13-15, H-1053 Budapest, Hungary Received 28 March 1990; accepted 28 November 1991 Compositio Mathematica 82: 355-372, 1992. (Ç) 1992 Kluwer Academic Publishers. Printed in the Netherlands. 1. Introduction Let k be a positive integer, L = log N, e(x) = e203C0ix, where A denotes the von Mangoldt function and IIp means product over prime numbers. A well known conjecture states that for any A &#x3E; 0. Conjecture (1) is still open, but several average versions of it are known to be true. For instance, using the circle method it is not difficult to prove that Huxley’s density theorem [5] implies that provided N1/6Lc H NL-c, c &#x3E; 0 suitable; see Heath-Brown [4] for a closely related result. Similarly, the Density Hypothesis in the form implies (2) for N03B5 H NL-c. Here N(a, T) denotes, as usual, the number of zeros 03C1 = 03B2 + i03B3 of the Riemann zeta function with 03B2 03C3 and |03B3|

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