Why would anyone have written about Jesus while He was alive, and who would have written about Him?
Galilee was the backwoods of the Roman Empire. No one cared what went on there.
And you do realize, don't you, that the only crucified person who is recorded by name in any Roman record is Jesus, right?
Also, people didn't write about things while they were happening, usually. For example, the formation of Rome wasn't recorded until hundreds of years after the fact.
Consider the times.
Besides, we don't even have all 1st century documents. Most have been lost.
And by the way, the Bible is not one book, but a collection of books, and all 27 books of the NT were not part of the Bible officially until the end of the 4th century, so they can be submitted as evidence.
Most historians believe Jesus existed. It's only a small fringe (of about a dozen "scholars") who hypothesize that Jesus is a myth.
Edit: As for the person who said that "everything" I wrote is "factually wrong", please demonstrate exactly how. Crucified people aren't mentioned by name in any Roman record (the only crucified person mentioned by name in any document is the Greek, Artacytus, in the 5th century B.C.), hardly any original documents from the 1st century have survived, and the NT really wasn't officially part of the Bible until the late 4th century. These are all historical facts.
Also, anyone who insists that writings from after Jesus' crucifixion are inadmissible as evidence is tantamount to insisting "photographs, or it didn't happen." Very little was recorded while it was happening, and then only if it held significance for many people or for the emperor.
What significance did Jesus have for the Roman Empire while He was alive? None whatsoever. There were other "rabble-rousers", and they were utterly insignificant. They were nothing, and that's what Jesus would have been considered, whether His miracles are a myth or not.