Gluon distribution at very small x from C-even quarkonia production at the LHC

C-parity-even quarkonia eta_b,c and chi_b,c with spin 0 and 2 are produced via two-gluon fusion. The expected cross section of the inclusive production of the quarkonia at the LHC, times the branching ratios of convenient decays, is up to tens of nanobarn per unit rapidity in the case of charmonia and around one nanobarn for the bottomonia. Measuring the quarkonia production as function of rapidity will allow to determine the gluon distribution function in nucleons in a very broad range of the Bjorken x from x~10^(-2) where it is already known, down to x~10^(-6) where it is totally unknown. The scale of the gluon distribution found from such measurements turns out to be rather low, Q^2 ~ 2.5 - 3 GeV^2, for charmonia and rather large, Q^2 ~ 20 GeV^2$, for bottomonia. We evaluate the scale by studying the next-to-leading-order production cross sections.