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cosec >students >Teaching >Winter 2008/2009 

Integers without large and small prime factors

Daniel Loebenberger (cosec - b-it)

Thursday 8 January 2009, 15.00, b-it  1.25 (cosec meeting room)

An integer n is called B-rough if it has no prime factor smaller than B. It is called C-smooth if it has only prime factors smaller than C. We consider the number of such integers smaller than a real positive bound x, in formulae:

πkB,C(x) := #{nx | n is a product of k primes in ]B,C] }

We essentially prove that this quantity is of order roughly

x / ln B

with an error of order roughly x / √B under Riemann hypothesis.

This result is inspired by an application in the number field sieve. As a byproduct, we can estimate the number of RSA-integers in a suitable sense.

Joint work with Michael Nüsken.

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