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The zeros of Euler’s Psi function and its derivatives. (English) Zbl 1112.33016

The aim of present paper is to study Euler’s Psi-function

ψ(x):=-1 x-γ- n=1 1 x+n-1 n,

with γ=0577 Euler’s constant, to investigate the locations of the points of inflexion and the positions of the stationary points of its derivative. The author works with the functions F 1 (x)=ψ(-x)+γ and

F k (x)= n=0 1 (x-n) k =-1 k-1F k-1 ' (x)=(-1) k-1 (k-1)!F 1 (k-1) (x)·

He gives some trigonometric approximations for the functions F 1 (x) and F 2 (x), which end to some bounds for the zeros of ψ. Finally, he investigates the properties of the horizontal distance between successive branches of the graph of F 1 and consequently ψ.

33E20Functions defined by series and integrals
30C15Zeros of polynomials, etc. (one complex variable)