MORE THAN 83.9% OF ZETA'S ZEROS ARE DISTINCT

The Riemann zeta function has infinitely many “non-trivial” zeros, points of the complex plane where it vanishes. Two questions about them remain open. Do they all lie on the critical line, where their real part equals one half? That is the Riemann hypothesis. And are they all simple — is each zero a single zero rather than two or more stacked at the same point? That is the simple zeros conjecture. According to the paper, nobody knows whether either of these statements implies the other.

Up to a height of 3 × 10¹², computers have checked that the zeros lie on the line and are simple. Beyond that, mathematicians prove proportions: at least such a share of all the zeros has the property.

A race measured in decimals

Kristian Muri Knausgård, in Kristiansand, Norway, sets out the recent history in detail. Proven shares of distinct zeros went from 63.9% to 70%, then jumped to 83.625% with an argument that the paper attributes to a preprint signed by Claude (Anthropic), verified by the mathematicians Alpöge and Furman, and reproved by Lamzouri. Small gains followed, including the author’s own previous paper at 83.6993%.

The paper’s introduction lists more than a dozen announced values for zeros that are simple and on the critical line, from 0.67300 to 0.673492, many posted in public code repositories between August and October 2026 and not refereed. The author states that these values have not been verified for the paper and that none of its results depends on them.

The new bounds

Without assuming the Riemann hypothesis or any other unproved statement, the paper proves that:

• more than 83.900% of the zeros are distinct (the exact bound is 1645064/1960733);
• consequently, more than 67.80% are simple;
• more than 67.353% are simple and on the critical line;
• at least 88.93% are simple or on the line, so zeros that are both off the line and repeated make up at most 11.07%.

Repeated zeros cost energy

The idea fits in one picture. Rescale the zeros so that they are spaced one unit apart on average, like beads on a thread. A theorem of Montgomery on how pairs of zeros are distributed fixes, asymptotically, a total “energy” summed over all pairs. A zero repeated d times contributes d² to this energy. Neighbouring zeros also use some of it, through their overlaps.

So if one can prove that neighbours necessarily use a lot of energy, little is left over for repeated zeros. The proof does this in three steps:

1. Zeros that keep a minimum distance from each other are handled exactly, thanks to an inequality known as the large sieve.
2. An inequality on seven consecutive zeros, proved with computer help, rewards double zeros more than simple ones. Its correction terms cancel out when added along the line.
3. A carefully designed test function, the “window”, makes this reward as large as possible.

The paper compares the problem to finding the lowest-energy state of a one-dimensional gas with two kinds of particles, simple and double zeros.

What the computer checked, and what it did not

The seven-point inequality required examining 60,467,309 boxes of possibilities, with no failure. The critical-line result needed more than 535 million. All main theorems are formalised in Lean 4, a proof-checking software, in about 13,300 and 10,000 lines on top of an existing library of about 102,000 lines, with statements about the zeros of zeta as defined in Lean’s mathematical library.

The author states one gap plainly. Each theorem rests on one extra axiom recording that a search program returned “true”. Lean’s kernel does not re-run that computation; it trusts Lean’s compiler and runtime for it. Three independent checks support the run, including identical counts from separate C++ and Python programs.

A method near its end

The paper also proves where its approach must stop. With its main window, it can never reach 84%: the ceiling is 0.83998. For any reasonably regular window, the ceiling is 0.84093 — less than 0.002 above the new result. Going further will need a different idea.

Conflict of interest. The author describes this work as “an experiment in AI-assisted mathematical research”. The author states that the arguments, computations, Lean formalisation, figures and much of the text were developed with substantial help from OpenAI Codex and Anthropic Claude, under the author’s direction; Claude developed the formalisation, the analysis of the method’s limits, the window and certificate of the critical-line theorem, the figures and much of the text, and both systems reviewed drafts “in the role of referees”. The starting argument is credited to a preprint signed by Claude. Claude also wrote the present article.