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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 22, 2017 10:10, Moscow, Steklov Mathematical Institute
 


Multiplicites of zeros of $\zeta(s)$ and its values over short intervals

A. Ivić

Serbian Academy of Sciences and Arts, Beograd
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A. Ivić
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Abstract: Let $r = m(\rho)$ denote the multiplicity of the complex zero $\rho = \beta + i\gamma$ of the Riemann zeta-function $\zeta(s)$. The present work, which is a continuation of [1], brings forth several results involving $m(\rho)$. It is seen that the problem can be reduced to the estimation of integrals of the zeta-function over “very short” intervals. This is related to the “Karatsuba conjectures” (see [2]), related to the quantity
$$ F(T,\Delta) := \max_{t\in[T,  T+\Delta]} |\zeta({\textstyle\frac12}+it)| \qquad 0 < \Delta = \Delta(T) \le 1. $$
By the complex integration technique, a new, explicit bound for $m(\beta+i\gamma)$ is also derived, which is relevant when $\beta$ is close to unity. As a corollary, it follows that, for $\tfrac{5}{6}\le\beta < 1$ and $\gamma\ge\gamma_1$, we have
$$ m(\beta+i\gamma) \le 4\log\log\gamma + 20(1-\beta)^{3/2}\log \gamma. $$

[1] A. Ivić, On the multiplicity of zeros of the zeta-function. Bulletin CXVIII de l'Académie Serbe des Sciences et des Arts – 1999, Classe des Sciences mathématiques et naturelles, Sciences mathématiques. №. 24. P. 119–131.
[2] A.A. Karatsuba, On lower bounds for the Riemann zeta-function. Dokl. Math. 63:1 (2001). P. 9 – 10 (translation from: Dokl. Akad. Nauk. 376:1 (2001). P. 15 – 16).

Language: English

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