Summary: Combining the arguments developed in the works of D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim [Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096), Acta Math. 204, No. 1, 1–47 (2010; Zbl 1207.11097)] and Y. Motohashi [Number theory in progress. Zakopane, Poland, 1997. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter, 1053–1064 (1999; Zbl 0941.11033)] we introduce a smoothing device to the sieve procedure of Goldston, Pintz, and Yildirim (see [Proc. Japan Acad., Ser. A 82, No. 4, 61–65 (2006; Zbl 1168.11041)] for its simplified version). Our assertions embodied in Lemmas 3 and 4 of this article imply that a natural extension of a prime number theorem of E. Bombieri, J. B. Friedlander and H. Iwaniec [Theorem 8 in Acta Math. 156, 203–251 (1986; Zbl 0588.10042)] should give rise infinitely often to bounded differences between primes, that is, a weaker form of the twin prime conjecture.