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) := #{n ≤ x | 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.