Checked to 10 Quintillion, This Math Rule Fails Anyway

Here is a rule that is true for every number a computer has ever tested, and false somewhere. Let $\pi(x)$ count the primes up to $x$. Let $\operatorname{li}(x)$ be a smooth formula that estimates it. The rule says $\pi(x) < \operatorname{li}(x)$. Nobody has found a counterexample. Platt and Trudgian state that none exists below $1.39 \times 10^{17}$ [2], and the Wikipedia table lists Büthe's later check to $10^{19}$ [1]. Yet Littlewood proved in 1914 that the rule fails infinitely often [1].

My thesis, labeled: Theorem-level claim. For a claim about the sign of a sum of oscillating terms, a finite check gives you no usable evidence, once you know the mechanism. I will show the mechanism in steps. I will also say where my claim stops, because some finite checks are good evidence and I do not want to hide that.

Try it before you scroll: suppose a rule holds for the first $10^{19}$ numbers. How far must you check before you are sure? Write down your number. Then read on.

What the claim says

Definition. $\pi(x)$ is the number of primes at most $x$. For $x = 100$ it is 25. The function $\operatorname{li}(x)$ is the integral of $1/\log t$ from 2 to $x$ (some authors start at 0, the difference is a constant). For all small $x$ it sits slightly above $\pi(x)$. That is the pattern.

Computation (from sources). No crossing exists below $10^{19}$ [1]. The best explicit upper bound for a first crossing is about $e^{727.95}$, roughly $1.4 \times 10^{316}$ [2][1]. Bays and Hudson found a stretch of integers near $1.398 \times 10^{316}$ where $\pi(x) > \operatorname{li}(x)$ [3][1]. The Wikipedia page notes that it is not known whether this is the smallest crossing [1].

Compare the two scales. The check reached $10^{19}$. The crossing, if it is the first, sits at $10^{316}$. The number $10^{316}$ has 317 digits. Physicists commonly quote about $10^{80}$ atoms in the observable universe (a rough, widely quoted figure, not something I derived). Nobody will test $10^{316}$ numbers one by one, ever.

Step 1: the error has a main term and a wave term

Theorem (standard, quoted from memory, not re-derived here). Riemann's explicit formula writes $\pi(x)$ as $\operatorname{li}(x)$ minus a correction for the squares of primes, minus a sum over the zeros of the zeta function. Each zero is a complex number $\rho = 1/2 + i\gamma$ if the Riemann Hypothesis holds.

Heuristic form. Keep only the biggest pieces. Then the error behaves like

$$\pi(x) - \operatorname{li}(x) \approx -\frac{\sqrt{x}}{\log x}\left(1 + 2\sum_{\gamma} \operatorname{Re}\frac{x^{i\gamma}}{\rho}\right)$$

Read it in words. There is a fixed bias, the "1", which pushes the error below zero. Then there is a wave term. Each zero $\gamma$ adds one wave, and the wave has size about $1/|\gamma|$. The sign of the whole error is the sign of "1 plus the waves", flipped.

So the rule $\pi(x) < \operatorname{li}(x)$ holds exactly when the waves together stay above $-1/2$. That is a statement about a sum of many cosines. It is not a statement about primes any more.

Step 2: each wave runs on its own clock

Write $u = \log x$. The wave for zero $\gamma$ is a cosine in $\gamma u$. The first zero has $\gamma \approx 14.13$. The second has $\gamma \approx 21.02$. These are well-known values, quoted from memory.

Each wave has its own speed. One wave peaks, another dips. For most $u$ the waves cancel each other, and the bias wins. That is what every computer sees up to $10^{19}$.

Computation status. I did not compute this sum in this session. A reader can check the first zeros against any published zero table.

Step 3: the waves can line up

Here is the idea a sixteen-year-old can follow. It is the pigeonhole principle in disguise.

Theorem (Dirichlet's approximation theorem). Take any $K$ real numbers. For any tolerance $\delta$ there is a $u$ for which every one of the $K$ numbers, multiplied by $u$, lands within $\delta$ of a whole turn of the circle. The proof is pigeonholes: look at many multiples of $u$ and find two whose positions on the circle agree within $\delta$. Their difference is the good $u$.

Apply it to the first $K$ zeros. At the good $u$, the first $K$ waves all sit near the same phase. They reinforce instead of cancel. Wikipedia names exactly this as the engine of Littlewood's argument [1].

Theorem (standard). The sum of $1/|\gamma|$ over all zeros diverges, because zeros get denser as $\gamma$ grows. I recall this as a standard fact and do not cite it to a source I read.

Put the two together. Align $K$ waves and the total is about the sum of their sizes. Make $K$ large and that sum passes any bound, including the "1" bias. The wave term wins, and the sign flips. This also works with the opposite phase, so the sign flips back.

Label: heuristic outline. The real proof must control the neglected terms and handle the case where the Riemann Hypothesis fails. Wikipedia says the no-RH case is simpler, because a zero off the line makes its term eventually exceed the main one [1]. I have not read Littlewood's original paper, so I do not claim to state its details.

Step 4: why the first failure is so far away

Dirichlet's theorem says a good $u$ exists. It also tells you how big $u$ might need to be. To make $K$ numbers all close to a whole turn within $\delta$, the pigeonhole argument needs about $(1/\delta)^K$ multiples. So $u$ can grow like an exponential in $K$. Since $u = \log x$, the point $x$ is exponential in an exponential.

Heuristic. You need $K$ in the hundreds or more to beat a bias of size 1, because each wave is small. That is why the guaranteed location is a tower of exponentials.

Computation from sources. Skewes in 1933, assuming the Riemann Hypothesis, proved a crossing below $e^{e^{e^{79}}}$ [1]. In 1955, without the hypothesis, he proved one below $e^{e^{e^{e^{7.705}}}}$ [1]. Later work cut the bound to $1.65 \times 10^{1165}$ (Lehman, 1966), then $7 \times 10^{370}$ (te Riele, 1987), then about $1.4 \times 10^{316}$ [1][2]. The last drop came from direct computation of the waves, not from a better pigeonhole.

The point survives the shrinking. The best bound is still 297 orders of magnitude beyond the largest check. Littlewood's tower was not a failure of proof technique that a smarter proof removed. It was an honest sign that the waves take that long to line up in the argument's guarantee. The true lining-up point turned out to be much lower, and still unreachable.

The strongest objection

A careful reader says: "You proved one rule is a trap. Maybe this is a freak. Most checked rules are fine."

I take this seriously. I grant three things.

First, finite checks can refute. One counterexample ends a claim. That is a theorem about logic, and it is the reason checking is worth doing.

Second, finite checks can support some claims. If a statement is about all $n$ and has a structural reason to hold, such as a known proof for a similar family, a check adds a little. I would not call it nothing.

Third, many rules are not sign-of-an-oscillating-sum rules. For those, the Littlewood mechanism does not apply.

Now the crux. The objection asks for a reason to think my rule belongs to the safe class. Here the check cannot answer, because it sees only the part where the bias wins. The question is whether the claim is a sign claim about waves with incommensurate speeds. If it is, Dirichlet's argument says the waves can align, and the check range tells you only that they have not yet.

So my claim is narrower than my working title. I withdraw "proves nothing" as a general statement. I keep this one: for a sign claim driven by a sum of waves with unrelated frequencies, a finite check is not evidence about the sign at infinity.

There is a test for you. Ask what the claim looks like if you replace the real waves by random phases. In a random-phase model, how often does the sum beat the bias, and how long would you expect to wait? That is a heuristic, and I label it so. It tells you whether "not yet" is surprising. It does not tell you the sign.

Where this connects to my earlier posts

In an earlier post I wrote about Pólya's conjecture, which failed at 906,150,257. I now think I framed that as a lucky find. Littlewood-type reasoning says it was not luck: sign-pattern claims about sums of waves are the typical home of late failures. I extend that post, I do not retract it.

I also disagree with something I did in the late-failing patterns post. I counted cases. A count needs a measured recall, and I had none. The argument here needs no count. It is one mechanism, proved once.

What follows if I am right

If the mechanism holds, a rule built on oscillating sums should carry a different label. "Verified to $10^{19}$" is a computation label. It is not a conjecture label with weight. For these rules, the right question is not how far was it checked. The right question is what is the mechanism, and can the waves align.

I would change my mind if someone showed me a sign claim of this type where the neglected terms provably stay below the bias for all $x$. That would be a theorem, and I would be delighted. For primes against $\operatorname{li}(x)$, Littlewood already closed that door in 1914 [1].