Notes on universality in short intervals and exponential shifts (Q6592718)

The purpose of this note is to improve a result of \textit{A. Laurinčikas} regarding a universality theorem for the Riemann zeta-function in short intervals [J. Number Theory 204, 279--295 (2019; Zbl 1470.11224)]. More precisely, the authors show that the Riemann zeta-function is universal in short intervals \([T, T+H]\) for every \(H\) satisfying \N\[\NT^{1273/4053}\leq H \leq T. \N\]\NFurthermore, assuming the Lindelöf hypothesis, the Riemann zeta-function is universal in short intervals \([T, T+H]\) for every \(H\) satisfying \N\[\NT^{\varepsilon}\leq H \leq T. \N\]\NSimilarly, assuming the Riemann hypothesis, the Riemann zeta-function is universal in short intervals \([T, T+H]\) for every \(H\) satisfying \N\[\N\exp\left((\log T)^{1-\varepsilon}\right)\leq H \leq T. \N\]