On the multiplicites of zeros of $\zeta (s)$ and its values over short intervals

Abstract

We investigate bounds for the multiplicities $m(\beta +i\gamma )$, where $\beta +i\gamma$ ($β\ge \1/2, γ>0)$ denotes complex zeros of $\zeta (s)$. It is seen that the problem can be reduced to the estimation of the integrals of the zeta-function over "very short" intervals. A new, explicit bound for $m(\beta +i\gamma )$ is also derived, which is relevant when $\beta$ is close to unity. The related Karatsuba conjectures are also discussed.

arXiv comment15 pages