New elementary estimates for the product of primes
by
Ed Bertram
University of Hawaii at Manoa

Let \theta(n) denote as usual the natural logarithm of the product of all the primes less than or equal to n. Very sharp inequalities relating \theta(p_n) and n(logn +loglogn -c), with 0 < c <= 1 and n >= N_c, have been obtained by G. Robin (1983). These rely on the classical results of J.B. Rosser, and L. Schoenfeld (between 1939 and 1976), and use analytic function theory, knowledge of the location of the non-trivial zeros of the Riemann zeta function, clever estimation and extensive computer calculations. The author will show (among other new elementary results), that there is one short, simple, elementary and direct proof technique resulting in good (and effective) upper and lower bounds, and uses the constants in Chebyshev-type estimates for \pi(n), or p_n.

Date received: March 7, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadx-57.