Three years after Paul's conversion Peter is still in and around Jerusalem [Acts 9 through 11] A.D.40.
Peter is imprisoned by Agrippa I and delivered by an Angel of the Lord [Acts 12]. Herod Agrippa dies [Acts 12:17-23]. This would have been the fourth year of the reign of Claudius according to Josephus....A.D.45.
A.D. 54 finds Paul visiting Peter again in Jerusalem [Galatians 2] and Peter returned the visit to Antioch and sees Paul [Galatians 2:11-14. The dates, of course, determined by [Galatians 2:1].
From A.D. 54 to A.D. 60 Peter apparently made journeys through the old Assyrian Empire (as instructed by Matthew 10:5-6) taking his wife along with him [I Corinthians 9:5].
When Paul salutes the folks in Rome in his epistle to them [Romans 16] in about the year 60 A.D., he never mentions the name of Peter. This would be quite strange if Peter had been holding an office in the Church there for some 25 years. He does mention 27 other people affiliated with the Church at Rome though.
While Paul is under arrest in Rome from A.D. 60/63 he writes four letters: Ephesians; Philipians; Colossians; and Philemon.....not once mentioning the presence of Peter. Right before his death Paul writes to Timothy stating that only Luke is with him [II Timothy 4:11] asking him to go get Mark (Peter's companion) and bring the scroll and Parchments that he had left at Troas [II Timothy 4:11-13]. Not once is Peter mentioned in this epistle and since Paul is asking for the presence of Mark only we can assume that Peter is already dead.
Peter, as an Apostle to the circumcised (Israelites), had no business in Rome......cause there were not any Israelites living there. They were Gentiles.
The assumption that Peter's imprisonment and angelic deliverance occurred in the same year as Herod's death is unjustified and unsupported.
he never mentions the name of Peter
It is likely that Paul knew that Peter was traveling away from Rome at that time. Given Peter's authority, only upon Peter's absence would Paul have written such an apostolic letter. Ray's timeline (mentioned earlier in this thread) shows Peter traveling in Bithynia, Pontus, and Cappadocia during this time (i.e. the time when Paul wrote the letter to the Romans).
Not once is Peter mentioned in this epistle and since Paul is asking for the presence of Mark only we can assume that Peter is already dead.
There are other alternatives to which you are not giving sufficient consideration. One is that Mark the nephew of Barnabas is not the "John Mark" of Acts 12 who wrote the Gospel of Mark and became the first bishop of Alexandria. Another is that since Timothy is in Ephesus, and Peter was in Rome, we shouldn't expect Paul to saying anything to Timothy about Peter. It is possible that Peter and Paul were both imprisoned at the same time, though in different cells. (I've actually visited the traditional cell in Rome where Paul is said to have been chained. It is utterly dark and dank, small, and not joined to other cells like in an American prison.) So we have to be careful not to put weight on arguments from silence when there are so other equally plausible explanations for Paul's not mentioning Peter. Peter himself refers to "our beloved brother Paul" in 2 Peter 3:15, and this epistle is thought to have been written shortly before Peter's death, since he says that the "laying aside of my earthly dwelling is imminent". (2 Peter 1:14). There is no good reason therefore, to reject the testimony of the patristic tradition that Peter and Paul were martyred at the same time under Nero in Rome, probably in 67 AD.
For those who think Peter was in Iraq (and not Rome), if you know anything about ancient geography, you will see that it would be very odd for him to be have been writing letters to "Pontus, Galatia, Cappadocia, Asia, and Bithynia" from present-day Iraq. (1 Peter 1:1; 2 Peter 3:1) Since he was writing letters to those places, he most likely had traveled there and in his two epistles was writing follow-up letters to strengthen the churches there in the faith. The only place relatively near to those locations, that would make sense to be referred to as 'Babylon', is Rome.
-A8