2026-08-29 · article Riemann hypothesisunsolved problemsthe explicit formula

The Riemann hypothesis — walking both ways between primes and zeros. And Λ = 0 exactly, which is to say no slack at all

Leanmachine-checked (Lean 4 + mathlib, standard axioms only, no sorryAx, no native_decide; theorem names given) paperproved, not yet machine-checked computationchecked on this machine, within the range stated; not a claim made to the outside knowna restatement, a known theorem, or a check of the literature

The Riemann hypothesis: the non-trivial zeros of the zeta function ζ(s) all lie on the line with real part 1/2.

A conjecture Riemann touched on in a single line of an 8-page paper in 1859. It is one of the Millennium Prize Problems, and it has gone unsolved for 167 years.

Why it is a problem about primes. The explicit formula Riemann wrote in the same paper is the reason.

The left side ψ(x) is the staircase of primes (the sum of log p over prime powers up to x). Not one prime appears on the right. What appears is ρ — the zeros of zeta, and nothing else.

The proportion of zeros — that 67.250% lie on the critical line is a theorem

This needs to be read carefully.
"67.250% of the zeros lie on the critical line" is a theorem. "The Riemann hypothesis is 67.250% proved" is not.
The former is a theorem about a different proposition, and a state of being 67% proved does not exist.

What is more, this 67.250% has reached 98.6% of the ceiling of 68.185% for pair correlation at bandwidth 1. Which is to say we are just short of the limit of the present tools.

It can be counted — and that is decisive

The Riemann hypothesis has a formula for how many zeros there are up to height T (Riemann–von Mangoldt). So one can establish that "if there are as many sign changes as that count, nothing has been missed in that range". Confirmed on this machine as well. computation

It can be written without complex numbers — Robin's criterion

The Riemann hypothesis has an equivalent form that uses no complex numbers at all.

σ(n) is the sum of divisors. A counterexample would be finite evidence — in logical terms it is Π₁. The margin is thin, though.

The interval for Λ — a quantity whose being pushed down is itself the proof

This is treated in detail in §06. Making 67.250% into 100% does not give the Riemann hypothesis, but pushing the upper bound on Λ down to 0 is the proof of the Riemann hypothesis itself. The 2026 candidates that have not been peer-reviewed (0.1875, 0.1787854, 0.172422), and how far the public logs alone reach, are in How far the published logs alone reach.

The starting point is a single fact.

A computation with complex numbers has 2 dimensions of input and 2 of output, 4 in all. So a brain optimised for three dimensions cannot process it as it stands.

This is not a metaphor but an exact description. The graph of a complex function lives in 4 dimensions and does not fit into 3-dimensional space. So mathematics has invented ways of dropping a dimension. We apply three common ones to the same function.

Escape 01: narrow the input to a single line and draw it over time

Fix the real part σ and move only the imaginary part t, and the input becomes 1-dimensional, so only the 2 dimensions of output are left and a curve can be drawn in the complex plane.

* The 0.00200 on the critical line is a residue from sampling t discretely; refine the step and it goes as close to 0 as you like.

What was thrown away: one dimension of input. The part of the hypothesis that says "it does not pass through on the other lines" cannot be checked in this picture.

Escape 02: push the 2 dimensions of output into colour

Hue = the argument of the output (a quantity that rotates, so it suits a cyclic hue), lightness = the magnitude of the output (compressed logarithmically). Then a single picture can be drawn over the input plane.

A zero is visible as a point where all the hues meet. Such points line up along a single vertical line — that is Re(s) = 1/2. The evenly spaced dark points at the left edge (σ < 0) are the trivial zeros (the negative even integers), which are not what the hypothesis is about.

What was thrown away: the readability of magnitude in the output. Hue is cyclic, so "which is larger" cannot be read.

Escape 03: on the critical line alone, the 4 dimensions become a 1-dimensional wave

The Riemann–Siegel Z function. This happens only on the critical line.

Four dimensions have fallen to a single real wave. The zeros can be counted by eye as "sign changes" — this is the substance of "it can be counted" in §02.

What was thrown away: the freedom to leave the critical line. Z is not real off the critical line. This escape can only be used inside "the world in which the hypothesis holds", and cannot be used to look for a place where the hypothesis breaks.

None of the three actually sees four dimensions

So what is 1/2?

1/2 is the position of a mirror. For a suitably normalised ξ(s), ξ(s) = ξ(1−s) holds, and the set of points left fixed by that exchange is Re(s) = 1/2 (the solutions of σ = 1−σ).

"It is symmetric" is the theorem; "it is on the axis" is the conjecture. Symmetry alone would allow zeros to exist as a pair at 0.3 and 0.7. The hypothesis says there is not one such pair.

The brain already does the same thing

And this constraint did not begin with four dimensions. What the eye receives is a 2-dimensional image, and the brain raises it into 3 dimensions from binocular disparity, from motion, and from the sense of distance in the hand.

Three dimensions are not seen directly either. Received after being dropped, then raised again from other cues — that is the normal state of perception, and four dimensions are not specially invisible.
What differs is the abundance of cues. The cues for raising three dimensions are built into the body; the cues for raising four have to be made oneself.

The explicit formula from §01, run on this machine. Both ways. computation

Forwards: add up the zeros and the staircase appears

Pairing ρ = ½ ± iγ, the contribution of one zero becomes a wave of frequency γ.

With 0 zeros the right side is just a straight line. Add waves to it and steps rise exactly at the primes. 2, 3, 4, 5, 7, 8, 9, 11 — at the primes and prime powers, and nowhere else.

The crests and troughs of the waves line up in the same direction at the positions of the primes. Everywhere else they cancel. The height γ of a zero is a frequency encoding the positions of the primes.

Backwards: get the heights of the zeros from the primes alone

If it is a spectrum, the inverse transform ought to work too. Using the prime powers up to 2 million, the following sum is computed. No information about the zeros was put in.

10 out of 10. At best they agree to the fourth decimal place.
That is to say the heights of the zeros are right there in the sequence of the primes.

* A window function (a Fejér-type triangular window) is needed. In the raw sum the peaks are buried in the background made by the edge of the truncation.

ψ(x) is "the sum over prime powers weighted by log p". This time it is π(x) — the number of primes up to x — itself.

First, count them ourselves

The values of π(10^k) are famous, and copying them out takes a moment. They are sieved for anyway. Copying from memory has produced errors elsewhere in this section (verifying labels). What can be counted, count. computation

R is far closer than li

R is 4 to 31 times closer than li. It is better by exactly as much as it anticipates the correction coming from the zeros.

And looking at the table, one more thing stands out. li − π is positive at every k.

A famous example of "what the numbers show" disagreeing with "what the theorem says". li being always above up to 10⁸ guarantees nothing at all.

Add up the zeros and the staircase of π(x) appears

The π side is far heavier than the ψ side. R is an infinite Möbius sum, and the exponential integral has to be computed at a complex argument every time. 200 zeros take 2,203 seconds. Even for the same theorem, which quantity you choose changes the weight of the computation.

This is not a metaphor. The "barely" part became a theorem in 2018.

Newman wrote in 1976: "even if the Riemann hypothesis is true, it is only barely true". Forty-two years later, Rodgers and Tao proved it.

What Λ is — seen in a small model

There is an operation of flowing heat into the zeros. Raise the time t and the zeros move, and at some time they all land on the real axis. That boundary is Λ.

The same happens with polynomials. Multiplying by e^(−t·d²/dx²) turns x²+a² into x²+a²−2t, and at t = a²/2 the complex roots become real.

The complex pair approaches the real axis and merges onto it at t ≈ 0.1020. This 0.1020 is the "Λ" of this polynomial. The Riemann hypothesis is "the Λ of zeta is at most 0" — that is, the claim that at t = 0 everything is already on the real axis. computation

The "barely" side became a theorem first

What was proved is the side that makes the problem harder. Had Λ < 0 there would be a safety margin and the road to a proof would have widened. Rodgers and Tao proved that no such margin exists. The way they did it is telling too: they use the "irregularity" of the zeros to deny the existence of slack.

Measuring that "barely"

Very close pairs are only 1/68 of what they would be if uncorrelated. The zeros repel one another. computation

The 1496th zero (at height 1977.17) has an index 1/229 of the median. Even at this modest height there is already a pair this close.

Λ is a gauge for the Riemann hypothesis itself

Proving Λ ≤ 0 is proving the Riemann hypothesis itself. Not a restatement — an equivalence.
The lower half is already closed. What remains is only "push the upper bound down to 0", and the moment that is done the Riemann hypothesis is proved.

This property is a rare one. "67.250% of the zeros lie on the critical line" does not automatically become the Riemann hypothesis when it reaches 100% (all the zeros being on the critical line and the proportion being 1 are different things). The upper bound on Λ is the only quantity whose being pushed down is itself the proof.

This section compares with the Collatz conjecture. The most important thing first.

On the single point of settlement, the two are in exactly the same place. Both are "unknown".
What the comparison below measures is not "the distance to a settlement" but "the amount and kind of knowledge accumulated".
And "more accumulated ⇒ settled sooner" is not guaranteed by history.

The first two rows are the goal, and there it is a dead heat. Only the third row differs. That third row splits into six axes.

Having accumulated more does not mean settling sooner

The two views from here on — "it cannot be crossed by accumulation" and "what can be proved is always the side with no slack" — turn up beyond this problem too. Five problems lined up against the same measure is Problems remain only at the exact edge.

One thing, though, that has the same shape

In both problems, what was proved is the "no slack" side.
For Riemann it is Λ ≥ 0 (no slack); for Collatz, q = 3 is the only criticality.
Mathematics has succeeded, for both, in showing that there is nowhere to run, and has not succeeded in producing an answer.

And one more thing. In both, the object is "as irregular as possible". The number of divisions in Collatz is perfectly equidistributed (Terras), and the zeros repel one another perfectly (1/68 of uncorrelated, measured in this article). The irregularity is a theorem; the regularity is a conjecture. Both stop there.

An estimate (an unmeasurable prediction about unmeasurable things)

The likelihood of truth is about the same. The difference is inside the resolution and means nothing. The numerical precedent (axis 05) is not a mark against Riemann — the Mertens conjecture is a strengthening of the Riemann hypothesis, and its failure says nothing about the Riemann hypothesis itself. If anything the marks against are on the Collatz side: the nearby 3n−1 has a cycle, and 5n+1 diverges.

Where the open items that have moved now stand is in What remains. Only what is open at present is placed here.

Why the two directions of Λ are asymmetric

The line by which Rodgers–Tao proved Λ ≥ 0 runs like this.

If Λ < 0, then at t = 0 the zeros would have to be more regular than they actually are.
But the zeros are not that regular — and that is already known. Hence Λ ≥ 0.

known "Why only one side?" has a name one step further on. What the proved side says is "the zeros are not equally spaced in local average" — a claim about a positive proportion of the zeros, within reach of tools that take averages. What the unproved side says is "not one zero lies off the critical line" — a claim about the absence of a zero-proportion defect. Zeros off the line, if there were any, could have density 0, and something of density 0 is not seen by any correlation, any moment, or any asymptotic formula. Knowing that the zeros are perfectly regular by every measure would not give the Riemann hypothesis. Averages see what is crowded and not what is sparse; only one side could be proved because only one side is a claim about the crowded part. Logical form does not explain the direction — Λ ≤ 0 is Π₁ and Λ ≥ 0 is Π₂, and the one higher in the hierarchy is the one that is proved. Collatz has the same shape: "almost every orbit attains an almost bounded value" is a positive proportion; "there is no cycle" is a single orbit.

known The quantities that can be measured on the zeros themselves are all restatements. At a zero ρ = 1/2 + iγ on the critical line, Re ζ′(ρ) > 0 ⟺ |S(γ)| < 1/2 (S the argument function; an identity stated verbatim in arXiv:2305.14253). Moving the zeros of the Hurwitz zeta ζ(s, a) in a, the first-order perturbation is dρ/da|_a=1 = ρζ(ρ+1)/ζ′(ρ) (elementary). Both restate analytic properties of ζ, and neither moves the direction of Λ.

One checkpoint. When a statistic measured on the zeros shows structure, first check whether a point process that does not use ζ (a uniform sequence of the same density, or the eigenvalues of GUE) gives the same value. If it does, it is an identity, not a discovery.

Sources and reproduction

There is no new mathematics in this article. The explicit formula is from 1859, the three visualisations are standard techniques, and Λ and Rodgers–Tao are known. What this article did was actually run it, produce the numbers, and make the pictures. The heaviest fact in the computations is that the backwards direction works — a peak rises in a sum over the primes, and its position differs from 21.022040 by only 0.0009.