The sum of multiplicative functions occurring in Selberg's sieve
by
George Greaves
University of Wales, Cardiff

The sum in the title is F_z(s) = \sum_{n < z^s, n|P(z)} g(n), where g is multiplicative and assumed to satisfy

å u <= p < v  g(p)logp < \kappalog(v/u)+O(1),

for some fixed ``dimension'' \kappa. We seek estimates of the form

Qz(s): = Fz(s)/Fz(\infty) >= \sigma(s)-Ez(s),

where \sigma(s) satisfies a certain integral equation and E_z(s) is to be an error term. A method will be outlined which can show slightly better than E_z(s) < Ce^-s/logz , for some constant C.

In the existing literature the estimate for E_z(s) increases with s, although it is known that both Q_z(s) and \sigma(s) are of the form 1+O(e^-s).

Date received: January 12, 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-12.