Effective Weak Universality in Short Intervals

Abstract

We prove an effective universality theorem of the Riemann zeta-function in short intervals $[T,T+H]$ with $T^{\frac{27}{82}}\le H\le T$ by following an effective multidimensional $\Omega$-result of Voronin. Furthermore, we also prove the results in short intervals $[T,T+H]$ with $T^{\epsilon }\le H\le T$ (for any fixed $\epsilon >0$) under the assumption of the Riemann Hypothesis.