The original version of this story appeared in Quanta Magazine.
Sometimes mathematicians try to tackle a problem head on, and sometimes they come at it sideways. That’s especially true when the mathematical stakes are high, as with the Riemann hypothesis, whose solution comes with a $1 million reward from the Clay Mathematics Institute. Its proof would give mathematicians much deeper certainty about how prime numbers are distributed, while also implying a host of other consequences—making it arguably the most important open question in math.
Mathematicians have no idea how to prove the Riemann hypothesis. But they can still get useful results just by showing that the number of possible exceptions to
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