Claude advances lower bound for Riemann zeta function to 67%

For 167 years, the Riemann Hypothesis has sat at the top of mathematics like an unconquered peak. Nobody has proven it. Nobody has disproven it. On August 10, Anthropic announced that an unreleased version of its Claude AI model just took a sledgehammer to one of those surrounding results. Claude increased the proven lower bound for the proportion of nontrivial zeros of the Riemann zeta function that lie on the critical line from 41.6% to 67.2%. That’s not a proof of the full hypothesis, which would require showing that 100% of those zeros behave as expected. But it’s the single largest improvement to this particular bound in the history of the problem. Why this number matters The Riemann Hypothesis, formulated by Bernhard Riemann in 1859, makes a precise claim about where certain special values of a mathematical function equal zero. Specifically, it says all nontrivial zeros of the zeta function have a real part of exactly 1/2, meaning they all sit on what mathematicians call the “critical line.” Since nobody has been able to prove all zeros land on the line, mathematicians have instead worked on proving that at least a certain percentage do. The previous best result, establishing...