Resolution of a conjecture on the convexity of zeta functions

Abstract

We settle a conjecture of Cerone and Dragomir on the concavity of the reciprocal of the Riemann zeta function on (1, infinity). It is further shown in general that reciprocals of a family of zeta functions arising from semigroups of integers are also concave on (1, infinity), thereby giving a positive answer to a question posed by Cerone and Dragomir on the existence of such zeta functions. As a consequence of our approach, weighted type Mertens sums over semigroups of integers are seen to be biased in favor of square-free integers with an odd number of prime factors. To strengthen the already known log-convexity property of Dirichlet series with positive coefficients, the geometric convexity of a large class of zeta functions is obtained and this in turn leads to generalizations of certain inequalities on the values of these functions due to Alzer, Cerone and Dragomir. (C) 2018 Elsevier Inc. All rights reserved.