On the Approximation of the Hardy Z-Function via High-Order Sections
Abstract
:1. Introduction
1.1. Riemann’s Analytic Extension of Zeta
1.2. The Approximate Functional Equation (AFE) and the Riemann–Siegel Formula
1.3. Sections of the Z-Function and Spira’s Approximation
- The number of terms required for Spira’s approximation, , increases quadratically compared to required by (4), making it far more costly for numerical calculations. This alone likely made (8), even if folklorically known, impractical for computational use, particularly before the advent of computers.
- The theoretical asymptotic estimation of Spira’s approximation error, obtained by classical means, is , which is similar to that of the Hardy–Littlewood formula, before the application of the Riemann–Siegel expansion of the error whose first term already gives an error of order . It should be noted, however, that these are approximate results, and hence, this seemingly superior asymptotic bound does not necessarily ensure greater practical accuracy over Spira’s approximation.
1.4. Spira’s Conjecture and the Absence of Theoretical Justification
2. Spira’s Approximation and Accelerated Approximations
3. Discussion and Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
- Riemann, B. “Über die Anzahl der Primzahlen unter einer gegebenen Grösse.” Monatsberichte der Berliner Akademie. In Gesammelte Werke; Teubner: Leipzig, Germany, 1892; Reprinted by Dover: New York, NY, USA, 1953. [Google Scholar]
- Hardy, G.H.; Littlewood, J.E. Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes. Acta Math. 1918, 41, 119–196. [Google Scholar] [CrossRef]
- Hardy, G.H.; Littlewood, J.E. The zeros of Riemann’s zeta function on the critical line. Math. Z. 1921, 10, 283–317. [Google Scholar] [CrossRef]
- Hardy, G.H.; Littlewood, J.E. The Approximate Functional Equation in the Theory of the Zeta Function, with an Application to the Divisor-Problems of Dirichlet and Piltz. Proc. Lond. Math. Soc. 1923, 21, 39–74. [Google Scholar] [CrossRef]
- Hardy, G.H.; Littlewood, J.E. The approximate functional equations for ζ(s) and ζ2(s). Proc. Lond. Math. Soc. 1929, 29, 81–97. [Google Scholar] [CrossRef]
- Edwards, H.M. Riemann’s Zeta Function; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Ivić, A. The Theory of Hardy’s Z-Function; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Siegel, C.L. Über Riemanns Nachlaß zur analytischen Zahlentheorie. In Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien; Springer: Berlin/Heidelberg, Germany, 1932; Volume 2, pp. 45–80, Also in Gesammelte Abhandlungen; Springer: Berlin/Heidelberg, Germany, 1966; Volume 1. [Google Scholar]
- Berry, M.V.; Keating, J.P. The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev. 1999, 41, 236–266. [Google Scholar] [CrossRef]
- Odlyzko, A.M.; Schönhage, A. Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Am. Math. Soc. 1988, 309, 797–809. [Google Scholar] [CrossRef]
- Gourdon, X. The 1013 First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height. 2004. Available online: http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf (accessed on 22 August 2024).
- Lehman, R.S. On the Distribution of Zeros of the Riemann Zeta Function. Proc. Lond. Math. Soc. 1970, 20, 303–320. [Google Scholar] [CrossRef]
- Odlyzko, A.M. On the distribution of spacings between zeros of the zeta function. Math. Comput. 1987, 48, 273–308. [Google Scholar] [CrossRef]
- Odlyzko, A.M. The 1020-th Zero of the Riemann Zeta Function and 175 Million of Its Neighbors. 1992. Unpublished Manuscript. Available online: https://www-users.cse.umn.edu/~odlyzko/unpublished/index.html (accessed on 22 August 2024).
- Titchmarsh, E.C. The Zeros of the Riemann Zeta-Function. Proc. R. Soc. London. Ser. Math. Phys. Sci. 1935, 151, 234–255. [Google Scholar] [CrossRef]
- Rosser, J.B.; Yohe, J.B.; Schoenfeld, L. Rigorous Computation and the Zeros of the Riemann Zeta-Function. In Congressional Proceedings of the International Federation for Information Processing; Spartan: Washington, DC, USA, 1969; pp. 70–76. [Google Scholar]
- Turing, A.M. Some calculations of the Riemann zeta-function. Proc. Lond. Math. Soc. Third Ser. 1953, 3, 99–117. [Google Scholar] [CrossRef]
- Platt, D.; Trudgian, T. The Riemann hypothesis is true up to 3·1012. In Bulletin of the London Mathematical Society; Wiley: London, UK, 2021. [Google Scholar] [CrossRef]
- Matiyasevich, Y. On some algebraic ways to calculate zeros of the Riemann zeta function. In Proceedings of the International Conference on Algebraic Informatics, Virtual Event, 27–29 October 2022; Springer: Cham, Switzerland, 2022; pp. 15–25. [Google Scholar]
- Beliakov, G.; Matiyasevich, Y. Approximation of Riemann’s zeta function by finite Dirichlet series: A multiprecision numerical approach. Exp. Math. 2015, 24, 150–161. [Google Scholar] [CrossRef]
- Cohen, H. Computational Number Theory, Past, Present, and Future. In Mathematics Going Forward; Morel, J.M., Teissier, B., Eds.; Lecture Notes in Mathematics; Springer: Cham, Switzerland, 2023; Volume 2313. [Google Scholar]
- Spira, R. Zeros of sections of the zeta function I. Math. Comput. 1966, 18, 542–550. [Google Scholar] [CrossRef]
- Spira, R. Zeros of approximate functional equations. Math. Comput. 1967, 21, 41–48. [Google Scholar] [CrossRef]
- Ivić, A. The Riemann Zeta-Function: Theory and Applications; Dover Publications: New York, NY, USA, 1985. [Google Scholar]
- Jerby, Y. An approximate functional equation for the Riemann zeta function with exponentially decaying error. J. Approx. Theory 2021, 265, 105551. [Google Scholar] [CrossRef]
- Paris, R.B. The asymptotic expansion of a sum appearing in an approximate functional equation for the Riemann zeta function. arXiv 2021, arXiv:2106.01371. [Google Scholar] [CrossRef]
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Jerby, Y. On the Approximation of the Hardy Z-Function via High-Order Sections. Axioms 2024, 13, 577. https://doi.org/10.3390/axioms13090577
Jerby Y. On the Approximation of the Hardy Z-Function via High-Order Sections. Axioms. 2024; 13(9):577. https://doi.org/10.3390/axioms13090577
Chicago/Turabian StyleJerby, Yochay. 2024. "On the Approximation of the Hardy Z-Function via High-Order Sections" Axioms 13, no. 9: 577. https://doi.org/10.3390/axioms13090577