Zero multiplicity and lower bound estimates of $|\zeta(s)|$

Abstract

We give an improved lower bound for $\max_{|T-t|\leq H} |\zeta(\tfrac{1}{2} + it)|$ when $2 \leq \alpha H \leq \log\log T - c$, $1 \leq \alpha \lt \pi$. Our theorem slightly refines the result in [11]. We also prove a theorem about an upper bound for the multiplicities of zeros of $\zeta(s)$ conditionally, assuming some lower bound for $\max_{|s - s_1| \leq \Delta} |\zeta(s)|$.