On Small Values of the Riemann Zeta-function on the Critical Line and Gaps between Zeros

0 |ζ( 1 2 + it)| 4 dt ∼ T 2π2 log T (T → ∞), this means that |ζ( 1 2 + it)| is small “most of the time”. The problem, then, is to evaluate asymptotically the measure of the subset of [0, T ] where |ζ( 1 2 + it)| is “small”. There are several ways in which one can proceed, and a natural way is the following one. Let c > 0 be a given constant, let μ(·) denote measure, and let Ac(T ) := {0 < t ≤ T : |ζ( 1 2 + it)| ≤ c}.