The inverse Riemann zeta function

• Autorzy: Kawalec A.

Identyfikatory

DOI

10.12921/cmst.2021.0000018

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

inverse Riemann zeta function; Euler prime product; non-trivial zero formula; Euler-Mascheroni constants; Stieltjes constants

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2021
• Tom: Vol. 27, No. 2
• Strony: 57--91
• Opis fizyczny: Bibliogr. 22 poz., rys.

Twórcy

autor

Kawalec A.

• 214 W Jennifer Ln, #6 Palatine, IL 60067, USA

Bibliografia

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Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).