The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros 1/2+iγ of the Riemann zeta function, we show that the sequence {equation*}Γ[a, b] =\{ γ : γ>0 and { log(| ζ^{(mγ)} (12+ i{γ }) | / (\!log{{γ }} )^{mγ})}{√{12loglog {γ }}} ∈ [a, b] \},{equation*} where the γ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a<b$ , and m_γ denotes the multiplicity of the zero 1/2+iγ . The same result holds when the γ ’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers γ (\!log T)/2π with γ ∈ Γ[a, b] and 0<γ ≤ T .