Forthcoming seminars
Seminar calendar
List of seminars
Archive by years
Register a seminar

Forthcoming seminars

You may need the following programs to see the files

General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences
May 17, 2012 14:00, St. Petersburg, POMI, room 311 (27 Fontanka)

Monotonicity of the Riemann zeta function and related functions

P. Zvengrowski

University of Calgary, Department of Mathematics and Statistics
Video records:
Windows Media 525.4 Mb
Flash Video 1,012.8 Mb
MP4 1,012.8 Mb

Number of views:
This page:477
Video files:123

P. Zvengrowski

Видео не загружается в Ваш браузер:
  1. Установите Adobe Flash Player    

  2. Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080
  3. Сообщите администратору портала о данной ошибке

Abstract: We shall consider monotonicity in the “horizontal direction” for several well known functions $f(s)$ of the complex variable $s = \sigma + it$, where monotonicity here means $|f(s)|$ is monotone increasing or monotone decreasing as $\sigma$ increases. The first function will be the well known gamma function $\Gamma(s)$, and it will be shown that $|\Gamma(s)|$ is monotone increasing in $\sigma$ once one is a small distance away from the real axis, more precisely for $|t| > 5/4$. A similar result will be shown for the Riemann zeta function $\zeta(s)$ as well as the two related functions $\eta(s)$ (the Euler–Dedekind eta function) and $\xi(s)$ (the Riemann $\xi$ function). Here it will be shown that all three have monotone decreasing modulus for $\sigma < 0$ and $|t| > 8$, and that for any of the three functions the extension of this monotonicity result to $\sigma < 1/2$ is equivalent to the Riemann Hypothesis. An inequality relating the monotonicity of all three functions will be given.

SHARE: FaceBook Twitter Livejournal
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018