Blazars, the most extreme subclass of Active Galactic Nuclei (AGNs), are observed at almost the full electromagnetic spectrum from radio to gamma-ray band. They get attentions by astronomers due to their violent activity. A number of telescopes also take them as important targets. The radiation from blazars is believed to be non-thermal emission of relativistic jets with small view angles to the line of sight. The emission from jets is Doppler boosted by beaming effect, and becomes extreme active. The research in blazars is very useful to jet physics, black hole, AGNs, and the high energy astrophysics, even the evolution of cosmology and galaxies. Thanks to the high sensitivity, broad energy range, and large view field of Large Area Telescope (LAT) on board the Fermi Gamma Ray Space Telescope, much larger sample and more accurate data can be achieved. It takes good chances, also challenges to blazar researches. For the theory research of blazar gamma-ray emission, as the radiation mechanism, the emission region size and location, the short timescale variability and variability origin, Fermi/LAT must be one of the most useful telescope at present. The radio observation has a very high spatial resolution and is easy to take long time monitors, so the connection between radio and gamma-ray emission can be used to constrain the radiation mechanism and the emission region of gamma-ray. Thus, we use the new data of Fermi/LAT and radio monitors to investigate the correlation of radio and gamma-ray band statistically. We use the cross sample of the second LAT AGN catalog (2LAC) and MOJAVE program. Then we collect the quasi-simultaneous data to investigate the correlation between radio and gamma-ray band, which is debatable before. Our results confirm the correlations of these two bands, but we find some new problems. The flux correlations of these two bands are affected strongly by selection effects. So the flux correlations should be used very carefully. Moreover, the correlations become worse when the gamma-ray energy bands get higher. This supports the leptonic model of gamma-ray emission. We also make the correlation tests of two important parameters in gamma-ray as photon spectral index and variability index with other parameters. We find some differences between FSRQs and BL Lacs in statistical results of variability index, which implies the variability behavior may differ below and above the high energy peak. The positive correlations of gamma-ray variability index and the gamma-radio luminosity ratio for FSRQs suggest that the variability should be from the gamma-ray emission region rather than the external radiation field. And the negative correlations of gamma-ray variability index and the gamma-ray photon spectrum index for FSRQs suggest the variability amplitude tends to be larger as the gamma-rays are closer to the high energy peak of SED. The negative correlations of the gamma-ray luminosity with gamma-ray photon spectral index suggest that the gamma-ray luminosity in Fermi/LAT range can not be used simply to investigate the blazar sequence instead of the the peak luminosity or the total luminosity, at least for FSRQs. However, our results are also limited by the source number of our sample. These results will be tested with larger samples. There are numbers of open questions in gamma-ray emission of blazars. The papers I read and the works I do are very limited during my master period. I hope I can do more in my following learning.
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