RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 Video Library Archive Most viewed videos Search RSS New in collection

You may need the following programs to see the files

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

 Video records: MP4 855.5 Mb MP4 217.1 Mb

Photo Gallery

 Видео не загружается в Ваш браузер: Активируйте JavaScript в Вашем браузере Установите Adobe Flash Player     Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080 Сообщите администратору портала о данной ошибке

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

 SHARE: