In the Annotated bible, one thing sprung out at me right away regarding the date of Jesus' birth. If you look at Luke, you end up with a later date than with the other Gospels. Luke says Mary was pregnant after Cyrenus started his first census. This would put his birth at a later date than when Herod was governor. The Annotated bible calls this a contradiction, and presumably say it is proof that Jesus either didn't exist or that the Gospels are wrong. They don't specify.
In order to draw that conclusion, you already have to assume that conclusion. It is circular reasoning. The conclusion is contained in the premise. The most that can be proven from the difference is that one date is wrong. Since humans are only born once that is all that can be proven. Historical dates are disputed all the time by scholars. When that happens, the dispute is noted, with reasons for and against. That is what neutral research does. No one then concludes that Napoleon or Julius Caesar didn't exist, or that everything written about Napoleon or Julius Caesar is wrong simply because theres two accounts of eithers birthdate or place. But they (the fine folks who contribute to the Annotated bible) do when it comes to Christ. Rather than following normal rules regarding academic neutrality in research, they use the facts in dispute to push an agenda. They are not neutral. To say that a difference in dates between two sources shows that all sources are invalid is just as much of a novice mistake as saying that both dates must be right because the Bible is infallible in every way and in all things. Both views are absurd, and both ignore the obvious simple conclusion that one date is an error, of which the authors may have never reached a compromise, or it was simply never omitted to maintain its journalistic accuracy Vs its historical accuracy.