Fe b 20 06 A Smoothed GPY Sieve

Combining the arguments developed in [2] and [7], we introduce a smoothing device to the sieve procedure [3] of D.A. Goldston, J. Pintz, and C.Y. Yıldırım (see [4] for its simplified version). Our assertions embodied in Lemmas 3 and 4 imply that an improvement of the prime number theorem of E. Bombieri, J.B. Friedlander and H. Iwaniec [1] should give rise infinitely often to bounded differences between primes. To this end, a rework of the main part of [7] is developed in Sections 2–3; thus the present article is essentially self-contained, except for the first section which is an excerpt from [4]. 1. Let N be a parameter increasing monotonically to infinity. There are four other basic parameters H, R, k, ℓ in our discussion; the last two are integers. We impose the following conditions to them: (1.1) H ≪ log N ≪ log R ≤ log N, and (1.2) 1 ≤ ℓ ≤ k ≪ 1. All implicit constants in the sequel are possibly dependent on k, ℓ at most; and besides, the symbol c stands for a positive constant with the same dependency, whose value may differ at each occurrence. It suffices to have (1.2), since our eventual aim is to look into the possibility to detect the bounded differences between primes with a certain modification of the GPY sieve. We surmise that such a modification might be obtained by introducing a smoothing device. The present article is, however, only to indicate that the GPY sieve admits indeed a smoothing; it is yet to be seen if this particular smoothing contributes to our eventual aim.