May 17, 2012. Peter Zvengrowski (University of Calgary). Monotonicity of the Riemann Zeta Function and Related Functions.

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.

List of talks at previous sessions of the seminar.