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Analytic Number Theory A Tribute to Gauss and Dirichlet Part 8

Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 132 D. A. GOLDSTON J. PINTZ AND C. Y. YILDIRIM by vp H the number of distinct residue classes modulo p occupied by the elements of H. The singular series associated with the . -tuple H is defined as 21 S H n 1 - 1 - VjpH . pp p Since vp H k for p h the product is convergent. The admissibility of H is equivalent to S H 0 and to vp H p for all primes. Hardy and Littlewood HL23 conjectured that 22 A n H A n h1 . A n hk N S H o 1 as N x. n N n N The prime number theorem is the k 1 case and for k 2 the conjecture remains unproved. This conjecture is trivially true if H is inadmissible . A simplified version of Goldston s argument in G92 was given in GY03 as follows. To obtain information on small gaps between primes let 23 ệ n h ệ n h -ệ n A m R n h AR m n m n h n m n h and consider the inequality 24 ệ n h - ệR n h 2 0. N n 2N The strength of this inequality depends on how well AR n approximates A n . On multiplying out the terms and using from G92 the formulas 25 AR n AR n k 0 k N A n AR n k 0 k N k 0 n N n N 26 AR n 2 Nlog R A n AR n N log R n N n N valid for k R N 2 log N -A gives taking h A log N with A c 1 27 n h - n 2 hNlog R Nh2 1 - o 1 A A2 - e N log N 2 N n 2N in obtaining this one needs the two-tuple case of Gallagher s singular series average given in 46 below which can be traced back to Hardy and Littlewood s and Bombieri and Davenport s work . If the interval n n h never contains more than one prime then the left-hand side of 27 is at most 28 log N V n h - n AN log N 2 N n 2N which contradicts 27 if A 2 and thus one obtains 29 lim inf Pn 1 - Pn 1. n x log pn 2 Later on Goldston et al. in FG96 FG99 G95 GY98 GY01 GYa applied this lower-bound method to various problems concerning the distribution THE PATH TO RECENT PROGRESS ON SMALL GAPS BETWEEN PRIMES 133 of primes and in GGOS00 to the pair correlation of zeros of the Riemann zetafunction. In most of these works the more delicate divisor sum 30 XR n r dp d r . r R d r n was employed especially because it led