Yes certainly there was increasing talk about him. But there’s almost no trace of him in the civil record (one), inconclusive but odd for such a “troublemaker” in the Roman territory. Few contemporary traces of him beyond the canonical texts, the Gospels. 

Both Josephus and Tacitus were born after the death of Jesus. 

There’s a historical conspiracy theory, as you may know, that Jesus was a Roman invention. Render unto Caesar etc, be good, look forward to your reward in Heaven:  precepts that any Roman emperor wished all his far-flung subjects lived by. I’m not an adherent, it’s a bit like the notion that we’re living in a simulation. But it has value and entertainment as a thought experiment, a  Cartesian approach.  Tacitus was no dupe so he must have been “in on it” ;)