On a Theorem of Matiyasevich

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Theorem of Matiyasevich B. Z. Moroz1, 2, 3* and A. A. Norkin4** 1

Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, 141701 Russia 2 St. Petersburg Division, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia 3 University of Bonn, Bonn, D-53012 Germany 4 Weizmann Institute of Science, Rehovot, 7610001 Israel Received March 1, 2020; in ﬁnal form, March 30, 2020; accepted April 15, 2020

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. DOI: 10.1134/S0001434620090047 Keywords: Riemann hypothesis, Diophantine equations, binomial coeﬃcients.

1. Any enumerable set is known to be Diophantine [1], [2]. Since the set of theorems of a formal mathematical theory is enumerable, it follows that the question of provability of a mathematical statement can be reduced to the question of solvability of a suitable Diophantine equation; cf. [3]–[6]. Following a procedure described in our papers [5], [6], one can construct, for example, polynomials f (x) and g(x), x := (x1 , . . . , xn ), n ∈ N, with integer rational coeﬃcients such that the statement (∃ a ∈ Zn ) f (a) = 0 (respectively, g(a) = 0) holds if and only if the Riemann hypothesis is formally provable (respectively, refutable) in axiomatic set theory. However, to write out these polynomials, several voluminous books would be required. On the other hand, as was noted in the survey [3] and the monograph [7], using the technique of Diophantine coding developed in the solution of Hilbert’s tenth problem, we can construct much simpler Diophantine equations whose unsolvability is equivalent to the Riemann hypothesis. Following [7, Sec. 6.4], Hernandez Caceres [8] was the ﬁrst to write out one such equation on several pages; certain inaccuracies contained in [8] were corrected in [9]. Making use of the restatement of the Riemann hypothesis proposed in a recent paper of Matiyasevich [10], one can construct much simpler systems of Diophantine equations the unsolvability of each of which is equivalent to the Riemann hypothesis. Such a system of equations was constructed in [11]; our present paper is a revised version of that paper. To make the exposition complete, we shall present a complete proof of Matiyasevich’s theorem, restoring some details of the proof omitted in [10] and slightly changing the notation used in Matiyasevich’s paper. Notation 1. As usual, ∅ is the empty set, Z is the ring of integers, R is the ﬁeld of real numbers, C is the ﬁeld of complex numbers, N is the set of natural (:=positive integer) numbers, N0 := N ∪ {0}, P is the set of primes, and Fp is the ﬁeld of residues modulo p for p ∈ P; we use boldface type to denote rows of elements of the same type, for example, u := (a, b, c), u ∈ N3 . By [a1 , . . . , an ] we denote the least common multiple of positive integers a1 , . . . , an ; the Chebyshev function ψ : N → R is deﬁn

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On a Theorem of Matiyasevich

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