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 n0 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]].