Abstract
Using the restatement of the Riemann hypothesis proposed in a recent paper of Matiyasevich, we explicitly write out the system of Diophantine equations whose unsolvability is equivalent to this hypothesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. V. Matiyasevich, “Diophantine representation of enumerable predicates,” Math. USSR-Izv. 5 (1), 1–28 (1971).
Yu. V. Matiyasevich, “The Diophantineness of enumerable sets,” Dokl. Akad. Nauk SSSR 191 (2), 279–282 (1970).
M. Davis, Yu. Matijasevič and Ju. Robinson, “Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution,” in Mathematical Developments Arising From Hilbert Problems (Amer. Math. Soc., Providence, RI, 1976), pp. 323–378.
V. H. Dyson, J. P. Jones and J. C. Shepherdson, “Some Diophantine forms of Gödel’s theorem,” Arch. Math. Logik Grundlag. 22 (1–2), 51–60 (1982).
M. Carl and B. Z. Moroz, “On a Diophantine representation of the predicate of provability,” J. Math. Sci. (N. Y.) 199 (1), 36–52 (2014).
M. Carl and B. Z. Moroz, “A polynomial encoding provability in pure mathematics (outline of an explicit construction),” Bull. Belg. Math. Soc. Simon Stevin 20 (1), 181–187 (2013).
Yu. V. Matiyasevich, Hilbert’s Tenth Problem (Nauka, Moscow, 1993) [in Russian].
J. M. Hernandez Caceres, The Riemann Hypothesis and Diophantine Equations, Master’s Thesis (Mathematical Institute, University of Bonn, Bonn, 2018).
B. Z. Moroz, The Riemann hypothesis and the Diophantine equations, Preprint no. 2018-03 (St. Petersburg Math. Soc., St. Petersburg, 2018) [in Russian].
Yu. V. Matiyasevich, “The Riemann hypothesis as the parity of special binomial coefficients,” Chebyshevskii Sb. 19 (3), 46–60 (2018).
A. A. Norkin, On Diophantine Equation whose Unsolvability is Equivalent to the Riemann Hypothesis, B. Sc. diploma paper (MPhTI, Moscow, 2019) [in Russian].
L. Schoenfeld, “Sharper bounds for the Chebyshev functions \(\psi(x)\) and \(\theta(x)\). II,” Math. Comp. 30, 337–360 (1976).
A. E. Ingham, The Distribution of Prime Numbers, Reprint of the 1932 original (Cambridge University Press, Cambridge, 1990).
N. Fine, “Binomial coefficients modulo a prime,” Amer. Math. Monthly 54, 589–592 (1947).
Yu. Matijasevič and Ju. Robinson, “Reduction of an arbitrary Diophantine equation to one in 13 unknowns,” Acta Arith. 27, 521–553 (1975).
M. Davis, “Hilbert’s tenth problem is unsolvable,” Amer. Math. Monthly 80, 233–269 (1973).
Acknowledgments
The authors are much obliged to the referee for a number of useful remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Moroz, B.Z., Norkin, A.A. On a Theorem of Matiyasevich. Math Notes 108, 344–355 (2020). https://doi.org/10.1134/S0001434620090047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620090047