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Zbl 1112.33016
Walker, Peter
The zeros of Euler's Psi function and its derivatives.
(English)
[J] J. Math. Anal. Appl. 332, No. 1, 607-616 (2007). ISSN 0022-247X

The aim of present paper is to study Euler's Psi-function $$\psi(x):=-\frac{1}{x}-\gamma-\sum_{n=1}^{\infty}\left(\frac{1}{x+n}-\frac{1}{n}\right),$$ with $\gamma=0577\cdots$ 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 $F_1(x)=\psi(-x)+\gamma$ and $$F_k(x)=\sum_{n=0}^{\infty}\frac{1}{(x-n)^k}=\frac{-1}{k-1}F'_{k-1}(x)=\frac{(-1)^{k-1}}{(k-1)!}F_1^{(k-1)}(x).$$ He gives some trigonometric approximations for the functions $F_1(x)$ and $F_2(x)$, which end to some bounds for the zeros of $\psi$. Finally, he investigates the properties of the horizontal distance between successive branches of the graph of $F_1$ and consequently $\psi$.
[Mehdi Hassani (Zanjan)]
MSC 2000:
*33E20 Functions defined by series and integrals
30C15 Zeros of polynomials, etc. (one complex variable)

Keywords: Euler's Psi function; location of zeros; separation of zeros

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