Smooth Beurling integers from irregular Beurling primes
by
Hugh Montgomery
University of Michigan

Roughly a century ago, Landau developed a method for proving the Prime Number Theorem by means of local lemmas. The information used about the distribution of the integers was only that [x] = x + O(1). In fact, a much weaker hypothesis, I(x) = cx + O(x^\theta), with \theta < 1, would suffice to give the Prime Number Theorem with the quantitative error term O(x*exp(-c\surd(logx))). In this way, Landau proved the Prime Ideal Theorem for algebraic number fields. We now show that Landau's method is best-possible in the sense that one can have a Beurling system with well-distributed integers, I(x) = cx + O(x^\theta), \theta < 1, but with the error term in the Prime Number Theorem \Omega(x*exp(-c\surd(logx))) if c is sufficiently large.

Date received: March 27, 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 # caew-57.