Here are two references to Jesus in Roman writings. In any case there is documentation that exists on Neros persecution of Christians in 43ad
This is a kind of evidence of Jesus existence, since people were willing to die rather than recant their faith, and that was within living memory of Jesus life.
From the annals of the Roman historian Tacitus. Annal 15:44
15.44.2. But, despite kindly influence, despite the leader's generous handouts, despite appeasing the gods, the scandal did not subside, rather the blaze came to be believed to be an official act. So, in order to quash the rumour, Nero blamed it on, and applied the cruellest punishments to, those sinners, whom ordinary people call Christians, hating them for their shameful behaviour.
15.44.3. The originator of this name, Christ, was sentenced to torture by Procurator Pontius Pilate, during the reign of Tiberius, but although checked for a moment, the deadly cult erupted again, not just in Judaea, the source of its evil, but even in Rome, where all the sins and scandals of the world gather and are glorified.
Pliny the Younger
Pliny was the governor of the Roman province of Bithynia, in present-day Turkey. In about 112 AD, he wrote (in Epistles X.96) to the emperor Trajan, asking for advice on how to deal with the Christians in his province, because he was executing so many of them. Pliny wrote:
'They were in the habit of meeting before dawn on a fixed day. They would recite in alternate verse a hymn to Christ as to a God, and would bind themselves by a solemn oath, not to do any criminal act, but rather that they would not commit any fraud, theft or adultery, nor betray any trust nor refuse to restore a deposit on demand. This done, they would disperse, and then they would meet again later to eat together (but the food was quite ordinary and harmless.)