  In this paper, we establish lower bounds on Weil height of algebraic integers
in terms of the low lying zeros of the Dedekind zeta-function. As a result, we
prove Lehmer's conjecture for certain infinite non-Galois extensions
conditional on GRH. We also introduce and study a condition on prime ideals
with small norms for arbitrary infinite extensions, in the spirit of a prime
splitting condition for infinite Galois extensions introduced by E. Bombieri
and U. Zannier.
