The Basel Problem

This famous conundrum, known as the Basel problem, is named after the hometown of both the Bernoullis and Euler, although it gained recognition far beyond Switzerland.

The central question is: What is the value of the infinite sum of the reciprocals of the squares?

1 + 1/2² + 1/3² + 1/4² + ...

as a closed-form expression? The ellipsis signifies that the series continues indefinitely.

Before revealing Euler's groundbreaking answer, it's essential to clarify what we mean by a closed-form expression. This concept can vary in interpretation among mathematicians. Generally, a closed-form expression is a mathematical formulation using a finite number of standard operations involving known constants and variables.

Thus, the Basel problem seeks to ascertain the total of this infinite series. Euler discovered that:

1 + 1/2² + 1/3² + 1/4² + ... = ?²/6.

This revelation was extraordinary. It's no surprise that Euler gained recognition for uncovering this mathematical gem. Not only does ? feature in the result, but it does so in a squared form, which is quite unusual. Typically, when ? appears in an equation, the associated circle or symmetry is evident; however, the presence of ?² raises intriguing questions.

Remarkably, a geometric explanation for this phenomenon exists, but that is a tale for another time.

Generalizations

After this monumental achievement, one might assume that Euler's work was complete, but he continued to investigate other infinite series with different exponents. He established that:

1 + 1/2^k + 1/3^k + 1/4^k + ... = ?(k)/90

and, in general, he derived a closed-form expression for all infinite sums of reciprocals of natural numbers raised to even powers.

Before delving deeper, we must adopt some notation. To avoid repeatedly writing out the beginning of the infinite series, we will utilize sigma notation.

For instance, the solution to the Basel problem can be represented as:

Similarly, the closed-form of the series of reciprocal quartics can be expressed in a comparable manner:

In general, these sums conform to a specific pattern. We will refer to these series as a function ? (pronounced “zeta”) of the exponent; thus, the Basel series can be denoted as ?(2), while the quartic series can be denoted as ?(4).

Euler discovered that if k is any positive integer, then:

The constants B are known as the Bernoulli numbers, which play a significant role throughout this mathematical theory. The studies conducted by the Bernoulli brothers ultimately contributed to understanding the problem that had long evaded them.

These numbers appear in various contexts, and while our comprehension of them is still limited, they are rational numbers and there are infinitely many of them, appearing frequently in number theory.

The sequence begins as follows: - B0 = 1 - B1 = ±1/2 - B2 = 1/6 - B3 = 0 - B4 = -1/30 - B5 = 0 - B6 = 1/42

Though they initially decrease in magnitude, they eventually become arbitrarily large. It is worth noting that for odd k greater than 1, Bk = 0. B1 is exceptional, and two conventions exist, so caution is warranted!

You can derive these values yourself on a leisurely day by calculating:

Here, the notation indicates that B1 = -1/2.

Thus, Euler didn't merely resolve the Basel problem; he also addressed infinitely many related problems using Jakob Bernoulli’s numbers in the process. Later, Euler revisited this challenge and provided multiple proofs. Initially, he employed properties of the sine function, and now numerous proofs exist.

However, this is not the complete narrative. By the 1800s, mathematicians had begun to explore complex analysis, and in the 1850s, the theory had matured enough to revisit the zeta function.

The Genius of Riemann

Bernhard Riemann was a German mathematician whose work spanned geometry and analysis, among other fields, but he had never considered publishing a paper in number theory. In 1859, Riemann released his first and only paper on this subject, forever altering the landscape of the field.

Riemann closely examined Euler’s zeta function and redefined it to accommodate complex numbers as arguments. This was a groundbreaking insight. In just a few pages, Riemann provided an explicit formula for prime numbers (the prime counting function) and a sketch of a proof for the prime number theorem, with a crucial detail centered on the zeta function.

At the turn of the 19th century, the prime number theorem was finally validated using Riemann's methodology and the zeta function. One of Riemann’s revelations was that the zeta function could be defined over a much broader domain than previously thought. By utilizing a method called analytic continuation, Riemann discovered a symmetric relationship between ?(s) and ?(1-s), enabling the calculation of zeta function values for negative arguments.

As a result, we now know the zeta values at negative integers. In general, ?(-2k) = 0 for k = 1, 2, 3, ... and more broadly:

You may have encountered the puzzling statement 1 + 2 + 3 + 4 + ... = -1/12 (which is, of course, misleading when presented this way). However, using the original definition of the zeta function ?(s) = ? 1/n^s, this infinite series is simply ?(-1), and applying the aforementioned formula with k = 1 reveals that since B2 = 1/6, we indeed arrive at -1/12.

Riemann then posed a question that continues to challenge mathematicians: the famous Riemann hypothesis concerning the location of the so-called non-trivial zeros of the zeta function. The zeta function is now referred to as the Riemann zeta function, honoring Riemann’s contributions.

The Hunt Has Only Just Begun

We now possess knowledge of the zeta function values at negative integers and positive even integers. Excellent!

Both Euler and Riemann recognized that the zeta function is undefined at the value 1, which is termed a singularity or pole. As Mengoli had previously demonstrated, the harmonic series diverges (1 + 1/2 + 1/3 + ... = ?), and since the zeta function evaluated at 1 is essentially the harmonic series, it too diverges.

In fact, even through analytic continuation, the zeta function remains undefined for s = 1, which is an established fact.

Thus, we understand the behavior at the odd positive integer 1, but Euler aimed to investigate the zeta values for all positive integers. However, even Euler encountered significant challenges in evaluating ?(3) and other odd integers.

During his work on the Basel problem, Euler was so adept at arithmetic that he likely had an inkling of the closed-form expression for ?(2) prior to proving it, which undoubtedly aided his efforts.

How could he have known?

He calculated the first 100 terms of the series, which approximate to about 1.6449. Recognizing this as ?²/6, one can only marvel at Euler’s brilliance.

He demonstrated several fascinating series relations and peculiarities, including unusual integrals, but he could not discover a closed form for ?(3), a dilemma faced by all mathematicians after him (if Euler couldn’t do it...).

Then in 1978, French mathematician Roger Apéry proved that ?(3) is irrational, meaning it cannot be expressed as a fraction of two integers.

This revelation was remarkable. We know little about this constant, and we lack a closed-form expression for it, while many other well-known constants remain elusive regarding their rationality. Yet, Apéry proved that ?(3) indeed is irrational.

His proof utilized an irrationality criterion from Dirichlet and involved complex series with specific relationships to ?(3). Shortly thereafter, numerous proofs emerged building on Apéry's insights, and the most straightforward ones are quite comprehensible for those with a Bachelor’s degree in mathematics or higher.

?(3) is sometimes referred to as Apéry’s constant in honor of his findings.

Closing Thoughts

To this day, no one has succeeded in finding a closed-form expression for Apéry’s constant. Each attempt seems to falter at the last moment due to cancellations or convoluted sums and integrals, which can be quite disheartening.

Some mathematicians have posited that this struggle stems from a simple truth: there may not be a closed-form expression for Apéry’s constant involving known constants. Or perhaps our axioms preclude us from obtaining such knowledge, similar to how the continuum hypothesis cannot be proven or disproven within ZFC axioms!

However, I prefer not to adopt this mindset. While I don't entirely dismiss it, I believe that with such an attitude, we will never uncover the truth if it exists. It's akin to declaring, “we will never reach Mars”—well, that’s certainly true if you don’t make the effort.

It may be necessary to break some conventions and embody what mathematician Edward Frenkel refers to as a “mathematical gangster,” much like Euler did, but we must endeavor to explore, and I suspect that innovative ideas are crucial.

We stand at the threshold of the quest for the zeta values of odd positive arguments, much like Bernoulli and Mengoli did when they were unable to determine ?(2).

What we need is another Euler!