Series involving the Zeta function and multiple Gamma functions

Estimating the digamma and trigamma functions by completely monotonicity arguments

“…In this paper, we study the complete monotonicity of the functions F a , then we apply these results to obtain new sharp bounds for the digamma and trigamma functions. The problem of estimating the gamma and polygamma functions has attracted the attention of many researchers, since they are close related to the theory of zeta functions [1,8,12,16,27,39], multiple gamma and related functions [7,[9][10][11]13,[24][25][26]30,33,35,38,40], gamma type distributions [15,29], or harmonic sums [23,34]. There are also many recent investigations dealing with one-sided and two-sided inequalities involving the digamma, trigamma, polygamma and other related functions, see e.g., [5,6,14,31,32].…”

A non-recursive formula for the higher derivatives of the Hurwitz zeta function

“…(2.39)], [15,34]), when −z ∈ N [16,1,7,3], for some special values of a [19], when a is rational [31], when (z) > 1 2 [9]. The first derivative of ζ has also been linked to some integrals involving cyclotomic polynomials and iterated logarithms in [1], polygamma functions of negative order in [2], the multiple gamma function in [8,3,4], and a log-gamma integral in [19] and in [7]. The second derivative with z = 0 has been studied in [13] (see also [26, p. 169] and [25, p. 651]).…”

Numerical computation of the Riemann zeta function and prime counting function by using Gauss–Hermite and Gauss–Laguerre quadratures

“…An efficient method for this purpose consists of using the integral forms of this function. There are many representation methods of the Riemann zeta function using the series [1,3,4,7,[10][11][12], and computations based on these series have been developed to evaluate this function in a real point, but the main question is, 'how many terms of these series must be computed to obtain a desired accuracy?' For example, for computing ζ(2.2) = ∞ n=1 1/n 2.2 in eight decimal digits of precision, 10 12 terms of the series must be computed and the time of the computing is high.…”