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 re...

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Bibliographic Details
Published inJournal of number theory Vol. 185; pp. 65 - 79
Main Author Ivić, Aleksandar
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2018
Subjects
Multiplicities of zeta-zeros
Riemann zeta-function
Short intervals
Multiplicities of zeta-zeros
Riemann zeta-function
Short intervals
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Summary: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.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2017.09.017