On irrationality criteria for the Ramanujan summation of certain series

Abstract

In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval [Formula: see text]. We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler–Mascheroni constant.