As usual let s = σ + it. For any fixed value t = t 0 with |t 0 | ≥ 8, and for σ ≤ 0, we show that |ζ(s)| is strictly monotone decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality relating the monotonicity of all three functions is proved: 1 2000 Mathematics Subject Classification : Primary 11M06, Secondary 11M26
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.