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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 theintegrals of the zeta-function over "very short" intervals. A new, explicitbound for $m(\beta+i\gamma)$ is also derived, which is relevant when $\beta$ isclose to unity. The related Karatsuba conjectures are also discussed.

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