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4 Computational Methods in Science and Technology

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Numerical Determination of a Certain Mathematical Constant Related to the Möbius Function

100%

Wolf M.

Computational Methods in Science and Technology

2023

tom Vol. 29, No. 1-4

17--20

We calculated numerically the value of some constant which can be regarded as an analogue of the EulerMascheroni constant

Asymptotic Properties of Stieltjes Constants

100%

Maślanka K.

Computational Methods in Science and Technology

2022

tom Vol. 28, No. 4

123--131

We present a new asymptotic formula for the Stieltjes constants which is both simpler and more accurate than several others published in the literature (see e.g. [1–3]). More importantly, it is also a good starting point for a detailed analysis of some surprising regularities in these important constants.

The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm

100%

Maślanka K. , Koleżyński A.

Computational Methods in Science and Technology

2022

tom Vol. 28, No. 2

47--57

We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80 000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 [19]. The crucial ingredient in this method is a sequence of high-precision numerical values of the Riemann zeta function computed in equally spaced real arguments, i.e. ζ(1 + ε), ζ(1 + 2ε), ζ(1 + 3ε), ... where ε is some real parameter. (Practical choice of ε is described in the main text.) Such values of zeta may be readily obtained using the PARI/GP program, which is especially suitable for this.

The inverse Riemann zeta function

88%

Kawalec A.

Computational Methods in Science and Technology

2021

tom Vol. 27, No. 2

57--91

In this article, we develop a formula for an inverse Riemann zeta function such that for w = ζ(s) we have s = ζ −1 (w) for real and complex domains s and w. The presented work is based on extending the analytical recurrence formulas for trivial and non-trivial zeros to solve an equation ζ(s) − w = 0 for a given w-domain using logarithmic differentiation and zeta recursive root extraction methods. We further explore formulas for trivial and non-trivial zeros of the Riemann zeta function in greater detail, and next, we introduce an expansion of the inverse zeta function by its singularities, study its properties and develop many identities that emerge from them. In the last part we extend the presented results as a general method for finding zeros and inverses of many other functions, such as the gamma function, the Bessel function of the first kind, or finite/infinite degree polynomials and rational functions, etc. We further compute all the presented formulas numerically to high precision and show that these formulas do indeed converge to the inverse of the Riemann zeta function and the related results. We also develop a fast algorithm to compute ζ −1 (w) for complex w.