

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)





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Monotonicity of the Riemann zeta function and related functions
P. Zvengrowski^{} ^{} University of Calgary, Department of Mathematics and Statistics

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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.

