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1
On a solution to a functional equation
Patkowski Alexander E.
Journal of Applied Analysis
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2022
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Vol. 28, nr 1
91--93
EN
We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi function.
2
Content available remote Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
Qi Feng
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
710--736
EN
In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.
3
Content available remote The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm
Maślanka Krzysztof, Koleżyński Andrzej
Computational Methods in Science and Technology
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2022
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Vol. 28, No. 2
47--57
EN
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.
4
Content available remote On a Relation Between Classical and Free Infinitely Divisible Transforms
Jurek Zbigniew J.
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
349--367
EN
We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.
5
Content available remote Are the Stieltjes constants irrational ? : Some computer experiments
Maślanka K. D., Wolf M.
Computational Methods in Science and Technology
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2020
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Vol. 26, No. 3
77--87
EN
Khnichin’s theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss-Kuzmin theorem to several thousand high precision (up to more than 53 000 significant digits) initial Stieltjes constants γn, n = = 0, 1, 2, . . . , 5000 in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We also study the normality of these important constants.
6
Content available remote Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line
Kawalec A.
Computational Methods in Science and Technology
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2019
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Vol. 25, No. 4
161--166
EN
In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with log(k). We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.
7
Content available remote Dirichlet series and gamma function associated with rational functions
Spreafico M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2010
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Vol. 58, no 3
189-195
EN
We investigate zeta regularized products of rational functions. As an application, we obtain the asymptotic expansion of the Euler Gamma function associated with a rational function.
8
Content available remote On the remarkable distributions of maxima of some fragments of the standard reflecting random walk and Brownian motion
Fujita T., Yor M.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 1
89--104
EN
In this paper, we consider some distributions of maxima of excursions and related variables for standard random walk and Brownian motion. We discuss the infinite divisibility properties of these distributions and calculate their Lévy measures. Lastly we discuss Chung's remark related with Riemann's zeta functional equation.
9
Content available remote Li's criterion for the Riemann hypothesis - numerical approach
Maślanka K.
Opuscula Mathematica
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2004
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Vol. 24, no. 1
103-114
EN
There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li [9]. The present paper reports on a numerical computation of the first 3300 of Li's coefficients which appear in this criterion. The main empirical observation is that these coefficients can be separated in two parts. One of these grows smoothly while the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true.
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