The zeros of Euler’s Psi function and its derivatives

We investigate the locations of the points of inflexion of Euler’s Psi function, and the positions of the stationary points of its derivative. We also establish some trigonometric approximations to Psi which lead to improved estimates for the positions of its zeros. Finally we consider the behaviour of the horizontal separation between the branches. © 2006 Elsevier Inc. All rights reserved.