The Riemann Hypothesis has many equivalent reformulations.
Some of them are arithmetical, that is, thewy are statements
about properties of integers or natural numbers.
Among them the reformulations with the simplest
logical structure are those from the class
$\Pi_1^0$ 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 $\Pi_1^0$ 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.
