The Mathematical Pattern Everyone Believes But No

There's something strange happening in mathematics. A pattern so compelling, so beautifully ordered, that virtually every mathematician believes it's true. They've tested it billions of times. They've built entire branches of mathematics on the assumption that it holds. But after more than 150 years, no one has managed to prove it.

This is the story of the Riemann Hypothesis, and it begins—oddly enough—with a thought experiment about infinitely shrinking rectangles.

When Infinity Doesn't Behave

Imagine lining up blocks in a row. The first block has width 1 and height 1. The second has the same width but half the height. The third has height one-third. The fourth, one-quarter. And you keep going forever, each block slightly shorter than the last.

Here's the question: what's the total area?

Your intuition probably says it should settle down to some finite number. After all, you're adding smaller and smaller pieces. Eventually they become so tiny they barely matter, right?

Wrong.

This sequence—known as the harmonic series (1 + 1/2 + 1/3 + 1/4 + ...)—doesn't converge. It grows without bound. Even though each individual rectangle keeps shrinking, the total area becomes infinite. You can prove this with a simple grouping trick: 1/3 + 1/4 is greater than 1/2. The next four terms also sum to more than 1/2. The next eight terms, more than 1/2. You keep adding halves forever, so the sum diverges.

As the Physics Explained video notes, "even though the rectangles shrink forever, their total area still becomes infinite."

So intuition fails. Mathematics corrects.

The Basel Problem: When Infinity Does Behave

But what if instead of adding inverse first powers (1/n), you add inverse squares (1/n²)?

This series shrinks much faster. At n=10, the harmonic series gives you 1/10, but the inverse squares give you 1/100. At n=100, it's 1/100 versus 1/10,000. The gap widens exponentially.

Does this converge?

Yes. The proof involves basic calculus—comparing the sum to the area under the curve y = 1/x². The integral from 1 to infinity of 1/x² equals 1, which bounds the series. So the sum of inverse squares must be less than 2.

But what exact value does it converge to?

This became the Basel Problem, posed in 1650 by Pietro Mengoli. Some of Europe's best mathematicians—including the Bernoulli brothers—tried and failed to solve it. Then in 1734, Leonhard Euler found the answer: π²/6.

The result stunned the mathematical world. An infinite sum of rational numbers converging to an exact expression involving π—a number that appears nowhere in the original problem's setup.

Euler didn't stop. He extended his method to inverse fourth powers, sixth powers, eighth powers. By 1739, he'd found the general pattern for all positive even powers. Each one had a beautiful closed form involving powers of π.

But the odd powers? Complete mystery.

The Stubborn Odd Powers

Euler tried to find clean formulas for the sum of inverse cubes, inverse fifth powers, inverse seventh powers. He failed. So did everyone after him.

It wasn't until 1978—nearly 250 years later—that Roger Apéry proved the sum of inverse cubes converges to an irrational number. A major breakthrough, but still no simple closed form like Euler's even powers.

The pattern strongly suggests these sums should have nice values somewhere between the even powers on either side. You can plot the even powers on a graph and practically read off where the odd powers ought to land. But "ought to" isn't proof.

So Euler did what mathematicians do when facing a family of related but partially understood problems: he unified them.

Enter the Zeta Function

Instead of treating each sum separately, Euler combined them into a single function: ζ(s) = 1/1^s + 1/2^s + 1/3^s + ...

When s=1, you get the harmonic series (divergent). When s=2, you get π²/6 (convergent). When s=4, you get π⁴/90 (convergent). The zeta function generalizes all of these into one object.

And here's where it gets interesting: for any value of s greater than 1, the zeta function converges. But for s less than or equal to 1, it diverges.

So it looks like the story ends there. The zeta function only makes sense for s > 1.

Except... maybe it doesn't.

The Function Beneath the Series

Consider a simpler infinite series: 1 + x + x² + x³ + ...

This geometric series only converges when -1 < x < 1. Outside that range, it blows up.

But through algebraic manipulation, you can show that within the convergent region, this series equals 1/(1-x). And here's the key: that function 1/(1-x) is perfectly well-defined for all values of x except x=1.

The series gives you a limited window. The function it represents exists more broadly.

As the video explains: "The series itself has a limited region of convergence, but the function it defines there may exist more broadly than the series that first revealed it."

Could the same be true for the zeta function?

Euler thought so. He introduced an alternating version—the eta function—where alternating signs create cancellation. This version converges for s > 0, not just s > 1. Then he showed how to express the zeta function in terms of the eta function, effectively extending zeta into a region where the original series makes no sense.

Euler hadn't changed the function. He'd revealed more of it.

This technique—called analytic continuation—would eventually become central to understanding what the Riemann Hypothesis is actually about. Because once you extend the zeta function beyond its original domain, strange patterns emerge in places you wouldn't expect.

Patterns involving the distribution of prime numbers. Patterns that seem to hold with perfect regularity across billions of test cases. Patterns that, if proven true, would unlock fundamental insights about the structure of mathematics itself.

But proving them? That's where the story gets really interesting.

— Nadia Marchetti, Unexplained Phenomena Correspondent