The inverse Riemann zeta function
214 W Jennifer Ln, #6
Palatine, IL 60067, USA
E-mail: art.kawalec@gmail.com
Received:
Received: 18 June 2021; revised: 27 June 2021; accepted: 28 June 2021; published online: 30 June 2021
DOI: 10.12921/cmst.2021.0000018
Abstract:
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.
Key words:
Euler prime product, Euler-Mascheroni and Stieltjes constants, inverse Riemann zeta function, non-trivial zero formula
References:
[1] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61, 11–13 (1979).
[2] S. Golomb, Formulas for the next prime, Pacific Journal of Mathematics 63 (1976).
[3] A. Kawalec, The nth + 1 prime limit formulas, arXiv: 1608.01671v2 (2016).
[4] A. Kawalec, The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function, arXiv: 2009.02640v2 (2020).
[5] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 9th printing, New York (1964).
[6] A. Kawalec, Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function, arXiv: 2012.06581v3 (2021).
[7] K. Knopp, Theory Of Functions Part I and Part II, Dover Publications, Mineola, New York (1996).
[8] R. Garunkštis, J. Steuding, On the roots of the equation ζ(s) = a, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 84, 1–15 (2014).
[9] G.N. Watson, A Treatise On The Theory of The Bessel Functions, Cambridge Mathematical Library, 2nd ed. (1995).
[10] D.H. Lehmer, The Sum of Like Powers of the Zeros of the Riemann Zeta Function, Mathematics of Computation 50(181), 265–273 (1988).
[11] Y. Matsuoka, A sequence associated with the zeros of the Riemann zeta function, Tsukuba J. Math. 10(2), 249–254 (1986).
[12] A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier 53, 665–699 (2003).
[13] A. Voros, Zeta Functions over Zeros of Zeta Functions, Springer (2010).
[14] The PARI Group, PARI/GP version 2.11.4, Univ. Bordeaux (2019).
[15] Wolfram Research, Inc., Mathematica version 12.0, Champaign, IL (2018).
[16] I.N. Sneddon, On some infinite series involving the zeros of Bessel functions of the first kind, Glasgow Mathematical Journal 4(3), 144–156 (1960).
[17] H.M. Edwards, Riemann’s Zeta Function, Dover Publications, Mineola, New York (1974).
[18] M. Coffey, Relations and positivity results for the derivatives of the Riemann ξ function, J. Comp. Appl. Math. 166, 525–534 (2004).
[19] M. Hassani, Explicit Approximation Of The Sums Over The Imaginary Part of The Non-Trivial Zeros of The Riemann Zeta Function, Applied Mathematics E-Notes 16, 109–116 (2016).
[20] R.P. Brent, D.J. Platt, T.S. Trudgian, Accurate estimation of sums over zeros of the Riemann zeta-function, Math. Comp. 90, 2923–2935 (2021).
[21] R.P. Brent, D.J. Platt, T.S. Trudgian, A harmonic sum over nontrivial zeros of the Riemann zeta-function, Bull. Austral. Math. Soc., 1–7 (2020).
[22] J. Arias De Reyna, Computation of the secondary zeta function, arXiv: 2006.04869 (2020).
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.
Key words:
Euler prime product, Euler-Mascheroni and Stieltjes constants, inverse Riemann zeta function, non-trivial zero formula
References:
[1] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61, 11–13 (1979).
[2] S. Golomb, Formulas for the next prime, Pacific Journal of Mathematics 63 (1976).
[3] A. Kawalec, The nth + 1 prime limit formulas, arXiv: 1608.01671v2 (2016).
[4] A. Kawalec, The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function, arXiv: 2009.02640v2 (2020).
[5] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 9th printing, New York (1964).
[6] A. Kawalec, Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function, arXiv: 2012.06581v3 (2021).
[7] K. Knopp, Theory Of Functions Part I and Part II, Dover Publications, Mineola, New York (1996).
[8] R. Garunkštis, J. Steuding, On the roots of the equation ζ(s) = a, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 84, 1–15 (2014).
[9] G.N. Watson, A Treatise On The Theory of The Bessel Functions, Cambridge Mathematical Library, 2nd ed. (1995).
[10] D.H. Lehmer, The Sum of Like Powers of the Zeros of the Riemann Zeta Function, Mathematics of Computation 50(181), 265–273 (1988).
[11] Y. Matsuoka, A sequence associated with the zeros of the Riemann zeta function, Tsukuba J. Math. 10(2), 249–254 (1986).
[12] A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier 53, 665–699 (2003).
[13] A. Voros, Zeta Functions over Zeros of Zeta Functions, Springer (2010).
[14] The PARI Group, PARI/GP version 2.11.4, Univ. Bordeaux (2019).
[15] Wolfram Research, Inc., Mathematica version 12.0, Champaign, IL (2018).
[16] I.N. Sneddon, On some infinite series involving the zeros of Bessel functions of the first kind, Glasgow Mathematical Journal 4(3), 144–156 (1960).
[17] H.M. Edwards, Riemann’s Zeta Function, Dover Publications, Mineola, New York (1974).
[18] M. Coffey, Relations and positivity results for the derivatives of the Riemann ξ function, J. Comp. Appl. Math. 166, 525–534 (2004).
[19] M. Hassani, Explicit Approximation Of The Sums Over The Imaginary Part of The Non-Trivial Zeros of The Riemann Zeta Function, Applied Mathematics E-Notes 16, 109–116 (2016).
[20] R.P. Brent, D.J. Platt, T.S. Trudgian, Accurate estimation of sums over zeros of the Riemann zeta-function, Math. Comp. 90, 2923–2935 (2021).
[21] R.P. Brent, D.J. Platt, T.S. Trudgian, A harmonic sum over nontrivial zeros of the Riemann zeta-function, Bull. Austral. Math. Soc., 1–7 (2020).
[22] J. Arias De Reyna, Computation of the secondary zeta function, arXiv: 2006.04869 (2020).
