Monotonicity results involving the zeta function with applications

Abstract

Let ζ be the zeta function. In this paper, we prove that the function ϕ(x,y) defined on (1,∞)2 byϕ(x,y)=2−xζ(x)−2−yζ(y)ζ(x)−ζ(y) if x≠y and ϕ(x,x)=(1−ζ(x)ζ′(x)ln⁡2)12x,is strictly increasing in both x and y on (1,∞). This gives a positive answer to a guess and an open problem. As applications, we obtain some new functional inequalities involving the ζ-function, and find a pair of simple but accurate lower and upper bounds for the ζ-function at odd integers.