Lower Bounds for the Number of Smooth Values of a Polynomial

Details Lower Bounds for the Number of Smooth Values of a Polynomial Lower Bounds for the Number of Smooth Values of a Polynomial

1998-07-20

arxiv.org/abs/math/9807102v2

Mathematics

Number Theory

24 pages. During the submission process, in communication with Cecile Dartyge and Gerald Tenenbaum a different approach to the

Scientific paper

We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.

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