The Fermi Large Area Telescope (LAT) revealed that blazars, representing the most extreme radio-loud active galactic nuclei (AGN) population, dominate the census of the gamma-ray sky, and a significant correlation was found between radio and gamma-ray emission in the 0.1-100 GeV energy range. However, the possible connection between radio and very high energy (VHE, E>0.1 TeV) emission still remains elusive, owing to the lack of a homogeneous coverage of the VHE sky. The main goal of this work is to quantify and assess the significance of a possible connection between the radio emission on parsec scale measured by the very long baseline interferometry (VLBI) and GeV-TeV gamma-ray emission in blazars, which is a central issue for understanding the blazar physics and the emission processes. We investigate the radio VLBI and high energy gamma-ray emission by using two large and unbiased AGN samples extracted from the first and second Fermi-LAT catalogs of hard gamma-ray sources detected above 10 GeV (1FHL) and 50 GeV (2FHL). For comparison, we perform the same correlation analysis by using the 0.1-300 GeV gamma-ray energy flux provided by the third Fermi-LAT source catalog. We find that the correlation strength and significance depend on the gamma-ray energy range with a different behavior among the blazar sub-classes. Overall, the radio and gamma-ray emission above 10 GeV turns out to be uncorrelated for the full samples and for all of the blazar sub-classes with the exception of high synchrotron peaked (HSP) objects, which show a strong and significant correlation. On the contrary, when 0.1-300 GeV gamma-ray energies are considered, a strong and significant correlation is found for the full blazar sample as well as for all of the blazar sub-classes. We interpret and explain this correlation behavior within the framework of the blazar spectral energy distribution properties.
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