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Let 0 < γ ≤ γ” denote the ordinates of consecutive nontrivial zeros of ζ(s) and set
In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let
where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let
where
and
where
Let
It is conjectured that for any k,
for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis implies
for all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).
