A person can live a righteous life, not sin, and yet not believe, and he'll still end up in hell. Asking someone to believe is like asking them to have a high IQ - either they do or they don't. If they don't, there's nothing they can do about it, except pretend to believe, and that I'm guessing wouldn't sit well with God either.
Christian salvation is based on faith not works. Good works follow from faith and Christians are expected to do good works.
Since "sin" is typically defined by each religion, and usually includes unbelief in their God as a sin, living without "sin" is impossible for a nonbeliever. But I understand your point is more generic.
There is a very strong correlation between good deeds and religious belief. CHaritable and non-profits have studied this at length, and you can find a good summary in Arthur Brooks book. Charities want to maximize the efficiency of fund raising and know that targeting religious populations is much more effective. People with a religious belief (not limited to Christianity) donate more, volunteer more, etc than people without religious belief.
It is unlikely that an atheist would live a rightous sinless life.
