Equidistribution of sequences associated with a-points of the derivatives of the Riemann zeta function
Hideki Murahara, Tomokazu Onozuka
TL;DR
The paper extends equidistribution results from the zeros of the Riemann zeta function to the $a$-points of its derivatives, establishing a general criterion: for any real-valued $C^2$ function $f$ satisfying six analytic conditions (C1)–(C6), the sequence $\{ f(\gamma_a^{(k)}) \}$ is uniformly distributed modulo $1$, where $\gamma_a^{(k)}$ are the imaginary parts of the $a$-points of $\zeta^{(k)}(s)$. The result is proved by expressing exponential sums over $a$-points via Stieltjes integration against the counting function $N_a^{(k)}(T)$, and applying Weyl's criterion together with oscillatory-integral estimates (via Lemma-type results from Titchmarsh). The work unifies unconditional results and the RH-improved regime, and includes a broad set of examples (e.g., $f(t)=u t^v (\log t)^w$, $f(t)=u (\log t)^v (\log\log t)^w$, and $f(t)=u (\log t)^v \zeta(t)$) demonstrating equidistribution for $a$-points of $\zeta$ and its derivatives. These findings generalize Fujii's framework and reveal the dense, regular fractional parts of advanced zeta-analytic sequences with wide applicability to analytic number theory.
Abstract
Fujii investigated the uniform distribution of various sequences associated with the non-trivial zeros of the Riemann zeta function by evaluating certain exponential sums over these zeros. In this paper, we present analogous results for a broader class of functions, establishing the uniform distribution of sequences arising from the $a$-points of the derivatives of the Riemann zeta function.