[This is a guest post by Caroline Turnage-Butterbaugh. This blog post was initially written in a different file format and converted using AI. — T.]
Evidence in favor of the Riemann Hypothesis comes in many flavors. OpenAI’s proof of the quasi-Riemann Hypothesis is definitely one such flavor; however, in this post I will focus my comments on a different kind of evidence: lower bounds on the proportion of nontrivial zeros of the Riemann zeta-function on the critical line. Great strides have recently been made on this problem, both with and without the use of AI. Many thanks to Terry Tao for inviting me to discuss this problem and the recent breakthrough results.
Background. Let , where are real. The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function have real part equal to 1/2. It arose out of Riemann’s foundational work, which led to proofs of the Prime Number Theorem by Hadamard and de la Vallée Poussin in 1896. If RH is true, then the error term in the Prime Number Theorem is essentially as small as possible.
Denote a nontrivial zero of by , where are real values and . The critical line is the line , and often we will refer to a nontrivial zero on the critical line as a critical zero. Riemann asserted that has infinitely many nontrivial zeros, and in 1914 Hardy proved that has infinitely many critical zeros. (See Chapter 10 of Titchmarsh for several proofs of this fact.) Platt and Trudgian have numerically verified RH up to , i.e. all of the nontrivial zeros for which have . (The nontrivial zeros are symmetric about the real line, so it suffices to consider those zeros for which .) The authors also show that these zeros are simple, which we expect to be true of every nontrivial zero. We set to denote the multiplicity of the nontrivial zero .
The recent proportionality results making headlines give a lower bound on the quantity Here, denotes the number of nontrivial zeros for which , and denotes the number of zeros counted by which are both simple and critical, i.e. for which and . One can analogously define to count critical zeros and set
Riemann–von Mangoldt formulagives a surprisingly precise asymptotic formula for ; for this discussion, it suffices to record that (Here and throughout, means and is a positive function.) Recall that zeros of analytic functions are counted by the argument principle with their multiplicity. Therefore:
*** It is important to keep in mind that we always count zeros with multiplicity.***
Previous key unconditional results on and . In 1942, Selberg proved that a positive proportion of the nontrivial zeros lie on the critical line. More specifically, he proved that there exists an effectively computable constant such that . (In his 1947 PhD thesis, S. H. Min calculated to be around .) In 1974, Levinson introduced a new method to show that , i.e. at least of the nontrivial zeros lie on the critical line. Soon after, he improved the proportion to . In 1979, Heath-Brown showed that Levinson’s method may be used to prove proportionality results on nontrivial zeros that are both simple and critical. Levinson’s method was extended by Conrey in 1989 to show that at least of the nontrivial zeros of are simple and critical. Over the years and with much effort and insight, members of the analytic number theory community have harnessed Levinson’s method to slightly improve these bounds, culminating with the work of Pratt, Robles, Zaharescu and Zeindler who proved in 2020 that at least of the nontrivial zeros of are on the critical line. The table below gives a snapshot of these results, with the precise proportions recorded.
| Author(s)/Year | | |
| Levinson (1975), Heath-Brown (1979) | | |
| Conrey (1989) | | |
| Pratt, Robles, Zaharescu, Zeindler (2020) | | |
A few key unconditional results on proportions of critical and simple zeros, prior to 2026.
Counting simple zeros under RH. In 1973, Montgomery published his groundbreaking work on the pair correlation of the zeros of . As a corollary to his main theorem, which we will state below, he showed that under RH at least of the nontrivial zeros are simple.
We now give a sketch of Montgomery’s argument for simple zeros. (To supplement the broad strokes which follow, I recommend Goldston’s expository paper.) This approach will lead us to the construction of Montgomery’s function , and, in turn, the statement of his theorem concerning the pair correlation of the zeros of .
Montgomery begins with the observation Note that since the argument assumes RH, the zeros are necessarily critical! The goal is to determine a constant such that so that, when combined with (2), we may conclude . A key transition occurs after noting (Keep in mind that we always count with multiplicity.) The sum on the right-hand side is a double sum over and , and with it pair correlation will soon enter the picture. Roughly, the idea is to find a nice function for which where and when . (Note that we have normalized the difference by , the average spacing between zeros around height .) For , define the Fourier transform by
[Note that and when . We may now state Montgomery’s Theorem.]
Theorem 1 (Montgomery, 1973) Assume the Riemann Hypothesis. Then is real, even, nonnegative, and, as , uniformly for .
(That the theorem holds up to is due to Goldston and Montgomery.) There are scores of interesting things to say about this theorem and Montgomery’s pair correlation conjecture, but we will stay on task and return to the simple zeros argument. Assembling the chain of inequalities above and adjusting to accommodate the weight , we find that Under RH, one can evaluate the RHS using Montgomery’s Theorem, provided and positivity is maintained. Inserting the Fejér kernel
(6)In other words, under RH at least of the nontrivial zeros are simple. Counting critical zeros via pair correlation under a weaker-than-RH assumption. It was known among experts for some time that Montgomery’s Theorem could be made unconditional. Indeed, the starting point of the proof of Montgomery’s explicit formula is an unconditional explicit formula due to Landau from 1909 (see equation (9) on page 353). Setting , where , the same arguments for Montgomery’s explicit formula and theorem carry through. That one could use an unconditional form of Montgomery’s Theorem to extract information about critical zeros was recently discovered by Siegfred Baluyot, Daniel Goldston, Ade Irma Suriajaya, and myself; henceforth BGST-B. In our first project we did not realize this capability of Montgomery’s method, and erroneously stated in the abstract “The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where .” In a follow-up paper we realized that, in fact, we could extract information on critical zeros using the pair correlation method, albeit upon an assumption weaker than RH. Roughly, we showed that if the nontrivial zeros of are contained in a narrow vertical box centered on the critical line, then the pair correlation method shows that at least 67.25% of the zeros are both simple and critical. I’ll give a sketch of the argument, highlighting some of the key distinctions from Montgomery’s work. Portions of what follows below appear in our preprint.
In addition to making use of the multiplicity of a nontrivial zero , we use a quantity we refer to as the horizontal multiplicity of . (This framework was suggested to us by Soundararajan and simplified our original classification of the ways in which can occur.) For a fixed real number, the horizontal multiplicity is the number of nontrivial zeros of on the horizontal line , counted with multiplicity. Assuming RH, is either zero or the multiplicity of the zero . Note that if , then on the line , there is one simple zero on the critical line.
Let denote the multiset of nontrivial zeros of for which . We can partition into disjoint multisets of zeros according to their horizontal multiplicity: Consider the multiset of simple zeros in and the multiset of critical zeros in . Furthermore, note that To obtain a lower bound on the proportion of simple and critical zeros, we begin with the following observation, which should be compared to
[(2)](#eqn-simple):
[Then, as in]
[(3)](#eqn-key), we transition to a pair correlation argument upon noting
[As in]
[(4)](#eqn-kernel_bound)we need to pick a kernel that maintains positivity.
Tsangfound a kernel that partially meets our needs; however, after rescaling for our setting, the positivity is only maintained provided the zeros are in a thin box centered on the critical line such that for some . Assuming this condition, we may use the unconditional version of Montgomery’s Theorem (see equations (3.2) and (3.3) in
[our preprint](https://arxiv.org/abs/2501.14545)) to bound from above the contribution of the right-hand side of
[(8)](#eqn-key2)in terms of the Tsang kernel. In particular, choosing yields .
Counting critical zeros via pair correlation unconditionally. On August 10, 2026, Anthropic announced that an unreleased research model of Claude had connected the BGST-B approach above with a 2000 paper by Bombieri to unconditionally prove that . The result was soon after verified by Alpöge and Furman, two mathematicians at Anthropic. Then, on September 2, 2026, Lamzouri announced a new, conceptually simpler proof of the same result. The entire preprint is very nicely written with helpful clarifications and remarks; in particular, Lamzouri takes care to compare/contrast his approach to that of Claude. With all of this in mind, I will focus this discussion on Lamzouri’s key ideas.
Lamzouri points out that the BGST-B approach cannot be made unconditional by merely finding a better kernel at the transition Lamzouri, in fact, avoids this transition altogether by changing the mechanism with which simple and critical zeros are identified. As he helpfully writes on page 5,
Our key idea is that one can bypass the inequality [(9)] and directly prove, for a large class of kernels , that the number of simple real elements of a finite multiset which is invariant under complex conjugation is without requiring that the individual off-diagonal terms be non-negative.
Instead of setting to be the multiset of nontrivial zeros for which as in BGST-B, he takes it to be the multiset
This is such a clever idea! This multiset is invariant under complex conjugation, i.e. if then and ; its real elements are those with corresponding zero satisfying ; its real and simple elements are those with corresponding zero being critical and simple. As Lamzouri notes in Remark 2.2, “the square [i.e., in ] is also naturally adapted to the Hilbert space argument in the proof [of Proposition 2.1] and to the use of Bessel’s inequality.” This new framework allows Lamzouri to bypass the portion of the previous arguments that required an assumption on the location of the nontrivial zeros. Concluding remarks. Since the breakthrough results by Anthropic and Lamzouri, subsequent minor improvements have been made. (I will refrain from quoting the strongest result at the time of writing, since it will likely be out of date by the time this appears!) It is important to keep in mind that even if we increased 67.25% to 100%, the Riemann Hypothesis would not be solved. (Refer back to the precise definition of which appears in (1).) Nevertheless, the old and new ideas which have come together to realize these breakthrough results have expanded our understanding not only of the nontrivial zeros of the Riemann zeta-function, but, more broadly, of how different techniques were used, pushed, and molded to breathe new life into this problem.