On the uniform distribution modulo $1$ of zeros and $a$-points of zeta functions

Abstract

Fujii gave five sufficient conditions for the uniform distribution modulo $1$ of the sequence $(u f(γ_n))$, where $γ_n$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. In this paper, we provide four sufficient conditions for the uniform distribution modulo $1$ that apply to a much broader class of sequences. Our method relies solely on the asymptotic behavior of the counting function and does not require the intricate calculations concerning the Riemann zeta function employed by Fujii in his paper. As applications, we prove the uniform distribution modulo $1$ of $(u f(x_n))$ for various sequences $(x_n)$, including the non-trivial $a$-points of the derivatives of the Riemann zeta function, the non-trivial zeros of the derivatives of Dirichlet $L$-functions, and the non-trivial zeros of functions in the Selberg class. Furthermore, by applying the Erdős--Turán inequality, we obtain an upper bound for the discrepancy of these sequences.