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

## TL;DR

This paper explores bounds on the multiplicities of zeros of the Riemann zeta function, linking the problem to integrals over short intervals and providing new explicit bounds especially near the critical line.

## Contribution

It introduces a new explicit bound for the multiplicities of zeta zeros and reduces the problem to estimating integrals over very short intervals.

## Key findings

- Derived a new explicit bound for zero multiplicities near the line .
- Reduced the problem to estimating integrals of  over very short intervals.
- Discussed implications for Karatsuba conjectures.

## Abstract

We investigate bounds for the multiplicities $m(\beta+i\gamma)$, where $\beta+i\gamma\,$ ($\beta\ge \1/2, \gamma>0)$ 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.