We study high-energy $gamma$-rays observed from Cyg X-3 by the Fermi Large Area Telescope and the 15-GHz emission observed by the Ryle Telescope and the Arcminute Microkelvin Imager. We measure the $gamma$-ray spectrum averaged over strong flares much more accurately than before, and find it well modelled by Compton scattering of stellar radiation by relativistic electrons with the power law index of $simeq$3.5 and a low-energy cutoff at the Lorentz factor of $sim!10^3$. We find a weaker spectrum in the soft spectral state, but only upper limits in the hard and intermediate states. We measure strong orbital modulation during the flaring state, well modelled by anisotropic Compton scattering of blackbody photons from the donor by jet relativistic electrons. We discover a weaker orbital modulation of the 15 GHz radio emission, which is well modelled by free-free absorption by the stellar wind. We then study cross-correlations between radio, $gamma$-ray and X-ray emissions. We find the cross-correlation between the radio and $gamma$-ray emissions peaks at a lag less than 1 d, while we detect a distinct radio lag of $sim$50 d with respect to the soft X-rays in the soft spectral state.
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