The Math Function That Looked Like A Coin Flip

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[00:00:00]In 1897, an Austrian mathematician named Franz Mertens was sitting at his desk in Vienna looking at a table of numbers.

[00:00:08]The numbers came from a function, a simple, almost boring-looking sum.

[00:00:14]And they did something he did not expect. They never grew very fast. They oscillated. They flipped between positive and negative. But they always stayed inside a narrow corridor, hugging zero, refusing to escape.

[00:00:29]Mertens did what mathematicians do. He guessed. He guessed that this function would stay inside that corridor forever.

[00:00:37]His guess had a stunning consequence. If it were true, it would immediately prove the Riemann hypothesis.

[00:00:43]The single most famous unsolved problem in mathematics. A problem the Clay Mathematics Institute would later put a $1 million bounty on.

[00:00:53]For almost a century, Mertens conjecture looked unkillable. People checked it for the first 10 million integers, then a hundred million, then a billion.

[00:01:03]Computers got faster, and the corridor held.

[00:01:07]Then, in 1985, two mathematicians killed it. They did not find a single integer where the conjecture fails. They proved, without ever exhibiting one, that the failures must exist.

[00:01:20]This is the story of how that happened, and of how a function built from the primes turned out to be just slightly more chaotic than anyone had dared to suspect. To understand Mertens, we need one ingredient, the Mobius function, written mu of n.

[00:01:37]It's a strange little function. It takes a positive integer and returns one of three values, plus one, minus one, or zero.

[00:01:47]Here is the rule. Factor n into primes.

[00:01:50]If any prime appears more than once, for example, 4 = 2 squared, or 12 = 2 squared times 3, return zero.

[00:02:00]If n is a product of distinct primes, count how many primes there are. An even number of distinct primes gives plus one.

[00:02:08]An odd number gives minus one. The number one gets plus one by convention.

[00:02:14]So, mu of one is plus one.

[00:02:16]Mu of two is minus one, one prime.

[00:02:19]Mu of three is minus one. Mu of four is zero. Mu of five is minus one.

[00:02:25]Mu of six equals 2 times 3, two distinct primes, plus one.

[00:02:31]Mu of seven, minus one. Mu of eight is zero, since eight contains 2 cubed.

[00:02:37]Mu of nine is zero.

[00:02:39]Mu of 10 is 2 times 5, two distinct primes, plus one.

[00:02:45]The Mobius function looks chaotic. It jumps. It vanishes whenever a square sneaks in. Roughly six out of every 10 integers give a non-zero value. Split almost evenly between plus and minus one.

[00:02:58]The other four out of 10 are killed by repeated primes.

[00:03:03]It looks like coin flips controlled by the primes, random, unpredictable. And it sits at the heart of analytic number theory, because the Mobius function is the inverse of the constant one function under a kind of multiplication called Dirichlet convolution. That fact links it directly to the Riemann zeta function, our dual function. Now, we build Mertens function, capital M of n.

[00:03:30]You start at zero, and you walk along the integers, adding mu of k at each step, plus one, then minus one, then minus one, then zero, then minus one, then plus one, then minus one, then zero, then minus one, then plus one, then minus one, then zero, then zero, then plus one.

[00:03:54]You're taking a random walk on the integers driven by the Mobius function.

[00:03:59]For the first few steps, M of N stays close to zero. It dips down. It climbs back up. It crosses zero many times.

[00:04:07]Let's look at specific values. M of 10 is minus one.

[00:04:11]M of 100 is one.

[00:04:13]M of 1,000 is two. M of 10,000 is minus 23.

[00:04:18]M of 100,000 is minus 48. M of a million is 212.

[00:04:24]M of 10 million is 1,037.

[00:04:27]M of 100 million is 1,928.

[00:04:31]The function is growing, but it is growing slowly.

[00:04:34]Compared to N itself, it is microscopic.

[00:04:38]M of a million is only about 200 against an N of a million. M of 100 million is under 2,000 against an N of 100 million.

[00:04:48]If the Mobius values really were independent coin flips, M of N would grow like the square root of N.

[00:04:54]That is just the central limit theorem applied to a sum of plus or minus ones.

[00:04:59]Square root of 1 million is 1,000.

[00:05:02]Mertens of 1 million is around 200, comfortably smaller. Square root of 100 million is 10,000.

[00:05:10]Mertens is under 2,000, still safely inside the corridor.

[00:05:15]Mertens looked at his table and made his conjecture, bold, specific. The absolute value of M of N is strictly less than the square root of N for every integer N bigger than one.

[00:05:28]The E seem. Why did anyone care about a random walk staying inside a corridor?

[00:05:34]Because of Bernhard Riemann.

[00:05:36]Riemann's zeta function and the Mobius function are linked by a beautiful identity. One over zeta of S equals the sum over N of mu of N divided by N to the s. This identity holds in the region where the real part of s is greater than one.

[00:05:52]And it extends by analytic continuation to a much larger region of the complex plane. The behavior of m of n controls the size of one over zeta, and that in turn controls where the zeros of zeta can sit. The exact theorem is this: If the absolute value of m of n is less than some constant c times the square root of n for all n for any positive constant c, then the Riemann hypothesis is true. Mertens' conjecture was the case c equals one, the cleanest, sharpest possible version.

[00:06:26]But any constant would have been enough.

[00:06:29]A bound of two times square root of n would have worked. A bound of 1,000 times square root of n would have worked. Just any constant at all. So, Mertens was not just guessing about an obscure function. He was offering a path to the holy grail. Compute one bound.

[00:06:46]Prove the Riemann hypothesis. Earn yourself an immortal name.

[00:06:51]No wonder people checked it. They checked by hand, then by mechanical calculator, then on punch cards, then on the first electronic computers, then on supercomputers. Every check, all the way out to the limits of what hardware could reach, came back the same.

[00:07:09]The conjecture held. It was the most heavily tested conjecture in number theory that nobody could prove. But the analysts had a problem.

[00:07:18]Random walks don't stay bounded.

[00:07:21]If the Mobius function really were like coin flips, then m of n would behave like an unbiased random walk on the integers. And there is a classical theorem about random walks, the law of the iterated logarithm. It says that a random walk with probability one will eventually exceed any constant multiple of square root of n infinitely often.

[00:07:43]The walk will not just leave the corridor. It will leave it again and again with no upper bound.

[00:07:50]So, if mu is genuinely random, Mertens is false, wildly false. The walk should not just escape the square root corridor. It should escape any constant times the square root corridor.

[00:08:03]But, maybe mu is not random. Maybe it has hidden structure.

[00:08:07]Maybe the primes secretly conspire to keep m small. People believed this sort of for a long time because the numerical evidence kept piling up.

[00:08:17]The conjecture refused to die.

[00:08:20]In 1942, mathematician Or Oystein showed something deeper. He proved that if Mertens conjecture is true, the imaginary parts of the zeros of zeta cannot be linearly independent over the rationals. They must obey an extra linear relation.

[00:08:37]That extra relation was suspicious. It looked artificial. It looked like the conjecture was asking the primes to be more disciplined than they had any reason to be.

[00:08:47]Most experts started to believe Mertens was probably false, but believing and proving are different.

[00:08:54]For another 43 years, the conjecture stood untouchable. The bound held in every numerical experiment. The argument that it should fail remained heuristic.

[00:09:06]Nobody could turn that suspicion into a theorem.

[00:09:10]Then in 1985, Andrew Odlyzko at Bell Labs and Herman te Riele in Amsterdam did something remarkable.

[00:09:18]They proved Mertens conjecture false.

[00:09:21]They did it without ever finding a specific n where it fails.

[00:09:25]Here is the idea. The error term in the partial sum m of n can be written by something called Perron's formula as a sum of contributions from the zeros of the Riemann zeta function. Each non-trivial zero contributes a wave.

[00:09:40]Each wave has a fixed amplitude that depends on the zero.

[00:09:44]Each wave has a frequency, and each wave has a phase that drifts as you move along the real line.

[00:09:51]If you stack many of these waves together with the right phases, they can constructively interfere.

[00:09:57]They can pile up.

[00:09:59]The sum of their contributions can grow without bound. Odlyzko and te Riele computed the first 2,000 non-trivial zeros of zeta to many decimal places of precision.

[00:10:10]They treated the amplitudes as coordinates of a vector in a 2,000-dimensional lattice. They asked, "Can we find a real number T such that all 2,000 of these waves line up close enough in phase to push the sum of their contributions above the value one?"

[00:10:27]Phrased this way, the question becomes a problem in lattice geometry. Are there short vectors in a particular lattice?

[00:10:34]Using a technique from theoretical computer science called LLL, the Lenstra-Lenstra-Lovasz lattice basis reduction algorithm, they answered yes.

[00:10:44]Their conclusion was a pair of inequalities. The limit superior of M of N divided by square root of N is at least 1.06.

[00:10:53]The limit inferior is at most -1.09.

[00:10:57]Both numbers are strictly outside Mertens' corridor of plus and minus one.

[00:11:02]The conjecture was dead, but here is the twist.

[00:11:06]They did not exhibit a specific integer N where M of N exceeds square root of N.

[00:11:11]They proved that one must exist somewhere.

[00:11:14]They did not say where. How big is somewhere?

[00:11:18]Pintz in 1987 gave the first explicit upper bound. He showed that the smallest counterexample lies below 10 raised to the power 3.21 * 10 to the 64.

[00:11:32]Later improvements have pushed the upper bound around quite a lot with different authors trading off arguments. Today, the best known upper bounds for the smallest n at which Mertens fails sits somewhere around 10 to the power 40.

[00:11:46]The exact value depends on which paper you trust.

[00:11:49]And the lower bound, direct computation has verified Mertens' conjecture for every integer n up to roughly 10 to the 16th power.

[00:11:58]Possibly further with more recent work, the smallest counterexample, if you wrote it out as a decimal number, has somewhere between 16 and 40 digits. We do not know exactly. A number that big is for all practical purposes invisible.

[00:12:14]The age of the universe in Planck units, the shortest meaningful unit of time, is roughly 10 to the 60. The number of atoms in the observable universe is around 10 to the 80. Even if you turned every atom into a tiny computer and let each one compute a billion previous values every second since the Big Bang, you would not finish checking every integer up to 10 to the 40. So, we have a counterexample we are mathematically certain exists and physically certain we will never see.

[00:12:43]A failure that is real but invisible.

[00:12:46]The first integer where Mertens was wrong is for any imaginable purpose an abstraction. We know its species. We know it lives somewhere in a vast cosmic neighborhood. We cannot get a name and address. Mertens' conjecture is one of mathematics' most famous near misses.

[00:13:03]It was beautiful. It was clean. It was simple to state. It would have solved the Riemann hypothesis in a single line of consequence. And it was wrong.

[00:13:12]The lesson cuts deep.

[00:13:14]Numerical evidence is treacherous in number theory. The first hundred million terms told one story.

[00:13:21]The truth, hiding somewhere past 10 to the 16th, tells another.

[00:13:27]The same pattern shows up across analytic number theory. Skewes number, where the prime counting function overtakes its predicted approximation at a value beyond 10 to the 300.

[00:13:38]Chebyshev's bias, where primes appear to prefer the residue three modulo four for stretches of trillions before flipping.

[00:13:45]The prime race. The Mertens conjecture.

[00:13:48]All of them feature regions where the numerical landscape looks one way until eventually far beyond anything humans can compute, it flips.

[00:13:57]What Odlyzko and te Riele showed is more important than the specific result.

[00:14:02]Proofs do not need explicit witnesses.

[00:14:05]You can rule out a conjecture by exhibiting structure in the zeros of zeta, by showing the waves must somewhere line up.

[00:14:13]The counterexample exists because the mathematics demands it. Not because anyone has ever seen it. This is a kind of proof by absent contradiction, and it is unsettling. It tells us that in number theory, what we have computed is a small bright patch in an otherwise dark landscape. The interesting behavior is almost always just beyond our reach.

[00:14:36]The Mobius function still looks like coin flips. The Mertens function still tracks a random walk that drifts slowly outward, exactly as the law of the iterated logarithm predicts. And the Riemann hypothesis still waits untouched. Mertens conjecture died, but the deeper question, how random are the primes really, is alive and well.

[00:14:59]The answer, when it comes, will probably not look like a bound on a sum. It will look like something we have not yet imagined.

[00:15:06]Until then, the primes keep their secret. And somewhere, far out beyond the visible mathematical universe, an integer waits, patient, enormous, undiscovered, for which Mertens was wrong.