St.Petersburg Department of Steklov Institute of Mathematics

This seminar is organized by St.Petersburg Division of Steklov Institute of Mathematics (POMI) of Russian Academy of Sciences.

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The next session(s) of the seminar.

The next (144^th) session will be held on Thursday, May 17, in the room no. 311 at 14.00.

We shall consider monotonicity in the "horizontal direction" for several well known functions f(s) of the complex variable s = σ+it, where monotonicity here means |f(s)| is monotone increasing or monotone decreasing as σ increases. The first function will be the well known gamma function Γ(s), and it will be shown that |Γ(s)| is monotone increasing in σ 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 ζ(s) as well as the two related functions η(s) (the Euler-Dedekind eta function) and ξ (s) (the Riemann ξ function). Here it will be shown that all three have monotone decreasing modulus for σ ≤ 0 and |t| > 8, and that for any of the three functions the extension of this monotonicity result to σ ≤ 1/2 is equivalent to the Riemann Hypothesis. An inequality relating the monotonicity of all three functions will be given.

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