Something fitting this description that I haven't seen mentioned here is Norman Levinson's proof that asymptotically 1/3 of the zeroes of the Riemann zeta function lie on the critical line, which was the best result of its kind at the time. He was a little over 60 when he proved this, shortly before his death. What I find most remarkable about this is that he didn't really do much number theory until his last few years.
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