Newsgroups: sci.math,sci.crypt
From: Dipankar Gupta <dg@hplb.hpl.hp.com>
Subject: Re: Papers on Smooth Numbers
Date: Tue, 16 Dec 1997 22:19:25 GMT

"Don Leclair" <dleclair@magma.ca> writes:

> Can anyone recommend any material specifically analyzing the
> properties of smooth numbers as they occur among general integers
> and quadratic residues?

A nice series of papers by NG de Bruijn used elementary methods to
look at smooth numbers.

N G de Bruijn, The asymptotic behaviour of a function occuring in the
theory of primes, J. Indian Math. Soc. (NS), 15 (1951), 25-32.

_____________, On the number of positive integers $\leq x$ and free of
prime factors $\gt y$, Nederl. Akad. Wetensch. Proc. Ser A. I. 54(1951),
50-60,  II 69(1966),239-247.

Halberstam (PLMS, 21(1970)) and Canfield, Erdös and Pomerance
(J. Number Th., 17(1983)) refined de Bruijn's lower bounds.

Also of interest is a series of papers by A. Hildebrand from 1983
onwards.

For a textbook treatment of analytic techniques, I would recommend a
read of chapter III.5 of Gérald Tenenbaum's text _Introduction to
analytic and probabilistic number theory_, Cambridge studies in
advanced mathematics, Tract #46, CUP (1995).

A historical survey of the asymptotic behaviour of Psi(x,y) is:

A. Hildebrand and G. Tenenbaum, Integers without large prime factors,
J. Théorie des Nombres de Bordeaux, 5 (1993), 411-484.

A neat proof of the prime number theorem using asymptotic properties
of Psi is given by:

H. Daboussi, Sur le théorème des nombres premiers, Comptes Rendus de
Acad. Sc. Paris, Série I, 298 #8, (1984), 161-164.

Regards,
Dipankar