  Using a generalized Littlewood theorem concerning integrals of the logarithm
of analytical functions, we have established a few equalities involving
integrals of the logarithm of the Riemann Zeta-function and have rigorously
proven that they are equivalent to the Riemann hypothesis. Separate
consideration for imaginary and real parts of these equalities, which deal
correspondingly with the integrals of the logarithm of the module of the
Riemann function and with the integrals of its argument is given. Preliminary
results of the numerical research performed using these equalities to test the
Riemann hypothesis are presented.
