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The Riemann hypothesis as the parity of special binomial coefficients. (Russian. English summary) Zbl 1434.11183

Chebyshevskiĭ Sb. 19, No. 3(67), 46-60 (2018); translation in Dokl. Math. 106, Suppl. 2, S256-S261 (2022).
Summary: The Riemann Hypothesis has many equivalent reformulations. Some of them are arithmetical, that is, they are statements about properties of integers or natural numbers. Among them the reformulations with the simplest logical structure are those from the class \(\prod_1^0\) from the arithmetical hierachy, that is, having the form “for every \(x_1,\dots,x_m\) relation \(A(x_1,\dots,x_m)\) holds”, where \(A\) is decidable. As an example one can take the reformulation of the Riemann Hypothsis as the assertion that certain Diophantine equation has no solution (such particular equation can be given explicitly).
While the logical structure of this reformulation is indeed very simple, all known methods for constructing such Diophantine equation produce equations occupying several pages. On the other hand, there are known other reformulation also belonging to class \(\prod_1^0\) but having rather short wording. As examples one can mention the criteria of the validity of the Riemann Hypothesis proposed by J.-L. Nicolas, by G. Robin, and by J. Lagarias. The shortcoming of these reformulations (as compared to Diophantine equations) consists in the usage of constants and funtions which are “more complicated” than integers and addition and multiplication sufficient for constructing Diophantine equations.
The paper presents a system of 9 conditions imposed on 9 variables. In order to state these conditions one needs only addition, multiplication, exponentiation (unary, with fixed base 2), congruences and remainders, inequalities, and binomial coefficient. The whole system can be written explicitly on a single sheet of paper. It is proved that the system is inconsistent if and only if the Riemann Hypothesis is true.

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11B65 Binomial coefficients; factorials; \(q\)-identities
11N37 Asymptotic results on arithmetic functions
11B68 Bernoulli and Euler numbers and polynomials
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