P. Erdős's problem on smooth numbers

Let P(t) denote the largest prime factor of the positive integer tge 2. Erdős conjectured that there is a constant c>0 such that if k is a sufficiently large integer and kle cn, then the inequality prod_{Sb n<tle n+kP(t)le kendSb}t<k!tag1 holds except for the case when for some t with n<tle n+k we have tge k! and P(t)le k. A. J. Hildebrand [Quart. J. Math. Oxford Ser. (2) {bf 36} (1985), no.~141, 57--69; [msn] MR0780350 (86f:11066) [/msn]] proved that for a fixed varepsilon>0 the inequality (1) holds for any integer nge c_0(varepsilon) and exp((1+varepsilon)log n/loglog n)le kle n^{1-varepsilon}. In the paper under review the author proves that for some positive constants c_1,c_2,c_3 the inequality (1) holds for any integer nge c_1 and exp(c_2(log n)^{2/3}(loglog n)^{1/3})le kle n^{c_3}. The proof uses a trigonometric sum estimate of A. A. Karacuba [Trudy Mat. Inst. Steklov. {bf 112} (1971), 241--255, 388. (errata insert); [msn] MR0330068 (48 #8407) [/msn]].