Those who deny the historical existence of Jesus have to deal with the Testamonium Flavianum. This is a passage the Antiquities of the Jews by Josephus that mentions Jesus.
About this time there lived Jesus, a wise man, if indeed one ought to call him a man. For he was one who performed surprising deeds and was a teacher of such people as accept the truth gladly. He won over many Jews and many of the Greeks. He was the Christ. And when, upon the accusation of the principal men among us, Pilate had condemned him to a cross, those who had first come to love him did not cease. He appeared to them spending a third day restored to life, for the prophets of God had foretold these things and a thousand other marvels about him. And the tribe of the Christians, so called after him, has still to this day not disappeared.
Most Mythicists reject the entire passage as a Christian interpolation. There is definitely reason to believe that a Christian tampered with what Josephus said as some of the passage comes across as too Christian.
There are three options in dealing with this passage: the passage is original as it stands, the entire passage is an addition, things were added to an original statement about Jesus.
Jesus Mythicists tend to reject the entire passage and then by extension, the passages about James the brother of Jesus and John the Baptist.
But is that the only way to look at it? I think a helpful parallel is the Johannine Comma. What is that? It is 1 John 5:7-8. Most modern translations present it in this way:
For there are three that testify: the Spirit and the water and the blood; and the three are in agreement.
But if you look in a King James Bible, you will find that same passage this way:
For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one.
Most scholars are agreed that the original passage about three witnesses provided a tempting opportunity to expand the passage into a reference to the Trinity. We (other than KJV-only Christians) can recognize that there was an original reference to the three witnesses that has been added to.
I think that is a likely parallel to what happened with the Testamonium Flavianum. Instead of creating a reference to Jesus out of nothing, someone added to (presumably trying to improve) the original reference to Jesus.
I doubt this will convince Jesus Mythicists but it is a helpful way for us to see what was going in this passage in Josephus.
