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.