Zeros of the digamma function and its BarnesG-function analogue

The zeros of the digamma function are known to be simple and real, but up to now few identities involving them have appeared in the literature. By establishing a Weierstrass infinite product for a particular regularization of the digamma function, we are able to find interesting formulas for the sums of the nth powers of the reciprocals of its zeros, for n≥2. We make a parallel study of the zeros of the logarithmic derivative of the Barnes G-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its Barnes G-function analogue.