A note on the zeros and local extrema of Digamma related functions

Abstract: Little is known about the zeros of the Digamma function. Establishing some Weierstrassian infinite product representations for a given regularization of the Digamma function we find interesting sums of its zeros. In addition, we study the same questions for the zeros of the logarithmic derivative of the Barnes $G$ function. At the end of the paper we provide rather accurate approximations of the hyperfactorial where, rather interestingly, the Lambert function appears.