On the multiplicities of zeros of ζ(s) and its values over short intervals

We investigate bounds for the multiplicities m(β+iγ), where β+iγ (β⩾12,γ>0) denotes complex zeros of ζ(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(β+iγ) is also derived, which is relevant when β is close to unity. The related Karatsuba conjectures are also discussed.