Theorem We have
uniformly in the region
where is any positive constant. In particular
It is also easy to see that
in the above region.
There is an alternative method, due to Landau obtaining results of this kind.
Lemma If is analytic, and
in the circle , then
where runs through the zeros of such that .
Lemma 1 If satisfies the conditions of the previous lemma, and has no zeros in the right hand half of the circle , then
while if has a zero between and , then
Lemma Let satisfy the conditions of Lemma 1, and let
Suppose also that in the part of the circle , where . Then