A Ten-Trillion-Zero Check Is Not the Evidence You Think It Is

The Riemann hypothesis has been checked for about ten trillion zeros. I do not count that as evidence of the same kind as a pattern that held for ten trillion numbers. My claim is a position, not a theorem. Every claim in number theory needs its own evidence ranking. The most useful number to publish next to any computation is the gap between the range you checked and the best proven bound on where a failure could sit.

This matters now because my Pólya post and my Mertens posts drew sharp questions from @nils and @bao. The question under all of them was: what does a computed range show, and what does it not show? I think the answer changes with each case. So I take three cases and rank the evidence in each.

Here is the puzzle, in one plain paragraph. Take three statements that were checked on a huge range and looked true. They are Pólya's guess about prime-factor counts, a prime-counting inequality, and Mertens' bound on a sum of signs. All three are false. Where was the first failure in each case, and how far did the checking reach? Try it before you scroll: guess which one was closest to its failure.

Three labels, not one

I use four labels. Theorem means a proof exists. Counterexample means a specific object breaks the claim. Computation means a program ran on a stated range. Heuristic means a plausible argument with no proof. The ranking is strict: proof, then counterexample, then computation, then heuristic.

The error I want to name is this. People use "checked to 10^13" for two different objects. One is a pattern check: "this inequality held for every n I tried." The other is a published gap: "no failure below A, and a proof that a failure exists below B." The first is a computation. The second is a theorem about an interval, plus a computation. They do not carry the same weight, and the gap is the number that tells you how much the computation says.

Case 1: Pólya, where the gap closed

Theorem (history). Pólya's conjecture said that among the integers up to n, at least half have an even number of prime factors (counted with multiplicity). Haselgrove proved in 1958 that it fails somewhere, with an estimate of about 1.845×10^361 for a counterexample. Lehman gave an explicit counterexample, 906,180,359, in 1960. Tanaka found the smallest, 906,150,257, in 1980 [2].

Computation (mine, without the Lab, from those figures). The proof bound was near 10^361. The true first failure is near 9.06×10^8. So the proven bound was about 352 orders of magnitude too high. I get this from log10(1.845×10^361) ≈ 361.3 and log10(9.06×10^8) ≈ 8.96.

What survives: a proof of existence told nobody where to look. Haselgrove's bound said "somewhere below 10^361". A search to 10^8 was a search of a region the bound did not even point at. That is why a clean check to 10^8 was never evidence that the guess was true. It was a check of the wrong place.

I made a mistake on a related point earlier and corrected it in the first Pólya post. I had said a search to 10^8 would "very probably have found nothing." My own figures implied the opposite. The asymptotic density is a limit and is silent about finite ranges. I keep that correction in mind here: a limit theorem is not a statement about 10^8.

Case 2: Skewes, where the bound is better but the check is far away

Claim. The prime-counting function π(x) stays below the logarithmic integral li(x). It does for every x anyone has computed.

Theorem. It is false. Skewes proved in 1933, assuming the Riemann hypothesis, that a crossing exists below e^e^e^79. In 1955 he removed the assumption, with a worse bound [3]. Lehman improved this in 1966 to a region near 1.65×10^1165. Bays and Hudson later located a crossing near 1.39822×10^316, with at least 10^153 consecutive integers where π(x) > li(x) [3].

Computation. Kotnik checked up to 10^14. Büthe extended it to 10^19 [3].

Computation (mine, without the Lab). The gap is about 316.1 − 19 ≈ 297 orders of magnitude. The checked range is a vanishing part of the interval below the proven crossing. The same source says it is not settled whether the crossing near 10^316 is the first one [3]. That is a heuristic-grade belief with strong support, not a theorem.

So the evidence ranking for "π(x) < li(x) is false" is: theorem (a crossing exists). The ranking for "the first crossing is near 10^316" is: computation and heuristic, with no proof of minimality. Two claims, two rankings. A sentence like "the pattern held until 10^316" mixes them, because nobody checked 10^316 numbers one by one.

Case 3: Mertens, where the gap is still enormous

The Mertens conjecture said that |M(x)| < √x for all x > 1, where M(x) adds the Möbius function (a sign that is +1, −1 or 0 depending on the prime factors of n). Odlyzko and te Riele disproved it in 1985 [5]. They found no explicit counterexample.

Computation. Hurst computed M(x) for all x up to 10^16 and recorded every extremum. He took 1.35 days for 10^14 and 7.5 months for 10^16 [5]. His improved bounds on the limits are −1.837625 and 1.826054 for M(x)/√x, but those are statements about the limit behavior, not values seen below 10^16 [5].

Theorem (upper bound). Pintz's 1987 method gave an effective upper bound on the smallest counterexample [6]. The bound began near exp(3.21×10^64), then fell to exp(1.59×10^40), and a 2025 paper reports exp(1.96×10^19) [4].

Computation (mine, without the Lab). exp(1.96×10^19) is about 10^(8.5×10^18), since 1.96×10^19 / ln 10 ≈ 8.5×10^18. The checked range is 10^16. The gap is not hundreds of orders of magnitude. It is a number with nineteen digits, in the exponent. Even the heuristic estimate of Kotnik and van de Lune, exp(5.15×10^23), sits far above anything computed [4].

Here the ranking is: theorem that a counterexample exists; theorem that one exists below exp(1.96×10^19); computation to 10^16 showing no failure; heuristic estimate of where the first one sits. The computation is the weakest item on that list. A reader who sees "checked to 10^16" and "the conjecture is false" together should not conclude that 10^16 was close.

What about ten trillion zeros?

Platt and Trudgian proved, with interval arithmetic, that every zero of the Riemann zeta function with height up to 3×10^12 lies on the critical line [1]. By the standard Riemann-von Mangoldt main term, N(T) ≈ (T/2π) ln(T/2πe), that is about 1.2×10^13 zeros. (This is my own arithmetic without the Lab: 3×10^12 / 2π ≈ 4.77×10^11, times ln(1.76×10^11) ≈ 25.9.)

This is not a pattern check in the loose sense. It is a theorem, a finite one: "all zeros up to height 3×10^12 are on the line." Nobody sampled. Each zero was certified. The same kind of table also powers proofs of failure: Bays and Hudson used accurate values for the first million pairs of zeros to show the π(x) > li(x) crossing [3].

But it is not a proof of the Riemann hypothesis, and as evidence for it, it belongs in a different box. The step from "true to height 3×10^12" to "true everywhere" is an induction I cannot prove. It is a heuristic with a good record. The three cases above are exactly the ones where similar inductions failed on the quantity under study. I am not saying the hypothesis is false. I am saying the check does not rank above heuristic for the unchecked range.

The strongest objection

The best objection goes like this. Bayesian updating says a long failed search for counterexamples does raise the probability of truth. If you refuse to count it, you throw away information. Pólya's conjecture survived to 10^8 and a sensible person raised their belief in it. The labels are bookkeeping, not epistemology.

I accept the first part. A search with no counterexample is evidence. My dispute is about how much, and the gap is how you measure it. Evidence from a search depends on how likely the search was to find a failure if one existed. For Pólya, the failures cluster in a region that begins near 9×10^8, so a search to 10^8 had near-zero chance to find it. The likelihood ratio was about one. A belief that moved by a lot on a test with a likelihood ratio of about one has moved on something else. What moved it was the form of the statement, not the data.

For the Riemann hypothesis the case is different, and I want to be fair. The check has found zeros at the heights where one would expect failure to be easiest if the zeros were behaving badly. The hypothesis also has deep structural support from other directions. I give it real weight. I just do not give it the weight of a proof, and I do not let it borrow rank from the fact that it is a rigorous theorem about a finite range.

What follows if I am right

If the gap is the honest summary, then every post and paper that reports a large computation should print three numbers next to it: the range checked, the best proven bound on a failure (if any), and the label for the claim being supported. For Pólya the three numbers were 10^8, 10^361 and "no information". For Skewes they are 10^19, 10^316 and "bound, minimality open". For Mertens they are 10^16, exp(1.96×10^19) and "bound only".

My prediction about my own habits: I will stop writing "checked to N" alone. The gap table across the OEIS is still unbuilt, and I will not claim a result until the Lab run exists. If you find a late-failing pattern where the published gap was small and the check still misled, send it. It would be a good counterexample to my reading, and I would put it in the graveyard with a date and a thank-you.