More than 83 percent of zeros of the Riemann zeta function are distinct
More than 83.69% of the zeros of the Riemann zeta function are distinct
Artificial IntelligenceLogic in Computer Science
Summary
The Riemann zeta function is a key object in mathematics connected to the distribution of prime numbers. Its zeros—special points where the function equals zero—are important to study, especially whether these zeros are distinct or repeated. This paper shows that at least about 83.7% of these zeros are unique, improving previous estimates. The authors use advanced mathematical tools and computer assistance, including formal proofs verified by software, to achieve this refined result.
What this means in practice
- •For cryptography developers: Improve confidence in assumptions about prime distributions that influence cryptographic algorithm security.
- •For mathematical software engineers: Enhance formal verification tools by integrating techniques for verifying complex inequalities and energy estimates.
Authors
Kristian Muri Knausgård
Abstract
The lower asymptotic proportion of distinct nontrivial zeros of the Riemann zeta function, counted with multiplicity, is at least $0.8369928814\ldots$. The previous bound was $0.83625\ldots$. As in the proof of that bound, an unconditional version of Montgomery's pair-correlation theorem gives an asymptotic energy estimate. The new ingredient is a short matrix inequality with a free clipping parameter. It strengthens the lower bound for this energy in terms of the number of distinct zeros. The gain is a nonnegative correction from overlaps between different nearby zeros on the critical line, which is retained even when some of these zeros are double. The matrix inequality, the threshold lemma, the block dichotomy, the counting assembly and the exact arithmetic are proved formally in Lean 4. The constant relies on a computer-assisted local inequality from recent work that has not yet been refereed. That computation was re-run independently, and every imported input is listed. This paper is primarily an experiment in AI-assisted mathematical research (Section 4).