Actions speak louder than words, they say.
In the years following Jesus' death, the persecution against Christians came primarily from the Jews as chronicled in the book of Acts (4:3-22, 5:17-42, 6:8-8:1, 8:3, 9:2, 9:23-24, 20:19, 23:12-14, 12:1-5, 13:44-51, 14:5-6, 14:19-20, 16:16-24, 17:1-15, 18:12-17, 19:23-41, 19:27-28:30). The interesting thing about this time period (from AD33 to AD60-ish) is there is no record of claims against the existence of Jesus. There are numerous arguments against claims made by Gnostic cults (1 John, 1 Peter, and others) and one possible argument made in defense of the resurrection (1 Cor 15:12-19). However, nowhere in the New Testament is the existence of Jesus defended.
By AD64, only 30-ish years after the crucifixion of Jesus, Nero began wide-spread persecution of Christians. During this time, countless hirtory books and ancient documents record Christians being imprisoned, tortured, and killed for their faith and their refusal to declare the emperor as god (and any other charges that could be trumped up against them such as setting Rome on fire). If, as some scholars believe, John wrote Revelation during the time of Domitian it could be a chronicling of the persecution of Christians by Rome. Polycarp, Irenaeus and other early Second Century church fathers also document the persecution during those times.
My point about the period of Roman persecution is this: There are not records of attempts to prove the Christians wrong. Rather than just killing them and thereby solidifying their faith, how much easier would it have been to issue proclamations that Jesus did not rise from the dead, or that He was living in Gaul with his prostitute wife, or that He never existed? This persecution began a whole generation after the death of Jesus and establishment of the church. Even a campaign of disinformation would have made sense. But even during this time that the eradication of Christianity became a priority of State we have no indicators that denials of any of the claims made by Christians concerning Jesus were false.
To me, this silence on the part of Rome and, before that, the Jewish authorities, is strong evidence.