Classical Results

Since the mid-14th century, it has been established that the following series,

known as the harmonic series, diverges and tends toward infinity as we incorporate more terms. This was demonstrated by the French philosopher Nicole Oresme around 1350. In fact, the partial sums increase at a rate similar to the natural logarithm—very slowly—but ultimately they reach infinity.

If we substitute the natural numbers in the denominators with prime numbers, the series will still diverge. However, replacing them with twin primes results in convergence, yielding a finite value, indicating that the density of twin primes among natural numbers is substantially lower than that of prime numbers.

Thus, while the harmonic series diverges, alternating the signs of the terms yields an intriguing outcome:

Here, ln denotes the natural logarithm (and if you're not familiar with it, don’t worry—you won’t need to grasp it to follow this discussion).

This serves as our initial indication that alternating signs can significantly alter the behavior of a series. We can readily demonstrate this using Taylor series for the natural logarithm.

Then, one might ponder: “What if we consider a similar alternating series but restrict ourselves to odd numbers in the denominators?”

At the dawn of the 14th century, the Indian mathematician Madhava of Sangamagrama uncovered the following remarkable result:

You may wonder why ? appears here.

Indeed, ? signifies that circles are involved in some capacity. This result can be derived from the Taylor expansion of the inverse tangent function, utilizing the fact that arctan(1) = ?/4, which connects it to circles.

This series bears the name of Gottfried Leibniz, who discovered it independently, albeit centuries later than Madhava.

The series is noteworthy for various reasons, but not due to its convergence properties, as it converges at an exceedingly slow rate.

“Calculating ? to 10 correct decimal places using direct summation of the series requires precisely five billion terms.” ~ Wikipedia.

So, what makes it significant?

This series, along with others we will encounter in this article, belongs to a broader family of series that are directly linked to prime numbers through a concept known as the Euler product, which we will discuss further along.

Now, we enter more fascinating territory. The aforementioned results are classical and viewed as elementary (not that they lack interest—they are indeed remarkable!), yet it turns out that raising the denominator to different powers leads to even more intriguing outcomes.

Three Centuries Later…

In the mid-17th century, mathematicians faced a challenge: express the following infinite series as a combination of known constants.

Here, the numbers in the denominators are square numbers. This challenge gained fame due to the numerous esteemed mathematicians who attempted and failed to solve it. A century of attempts passed before a young and relatively unknown mathematician named Leonhard Euler decided to tackle the problem.

In a moment of insight, Euler discovered a connection between this daunting mathematical challenge and a well-known function that mathematicians had been exploring for about a thousand years: the sine function.

Although modern trigonometry originated in India during the 5th century, no one had realized that the sine function could be expressed as an infinite product of simple factors. Euler made this discovery, which proved to be the key to solving the puzzle.

In 1734, Euler used this insight to demonstrate that:

This result is still regarded as one of the most surprising and beautiful findings in mathematics, and Euler continued to prove it in various ways to gain a deeper understanding of the theorem. This outcome is known as the Basel problem, named after Euler's hometown.

His proof gained worldwide recognition for its ingenuity and brilliance, marking a significant intersection between the vast fields of analysis and number theory.

You may wonder where number theory fits in. It took another brilliant mind to fully realize the connection between these infinite series and prime numbers, but Euler was the first to indicate that there is indeed a relationship, which we will explore in a future article.

What about the alternating version of this series? Once Euler's result was established, it became straightforward to demonstrate that

Euler continued to investigate higher powers, yielding astonishing results that paradoxically depend on certain rational numbers named after Jakob Bernoulli, who initially failed to solve the Basel problem.

For instance, when the power is 4:

In fact, he derived a general formula for such series whenever the denominators are even powers.

The Mystery

Through the use of Fourier analysis, we can establish that

This result is remarkable in its own right and is somewhat akin to Leibniz’s series for ?, as it also alternates and has odd numbers as bases in the denominator powers.

At this juncture, we have gathered substantial evidence. To summarize, here is what we have examined thus far:

However, an astute reader may notice some gaps in our puzzle. The first missing piece is the evaluation of the series:

This number is referred to as Catalan’s constant, named after mathematician Eugène Catalan, who published a memoir on it in 1865.

This series appears just as straightforward as the preceding ones, but to this day, the challenge of finding a closed-form solution remains unresolved! Moreover, we do not even know if G is rational or irrational—specifically, can G be expressed as a fraction of two integers?

G has been labeled:

“Arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven.” ~ Notices of the American Mathematical Society, 60 (7): 844–854.

We know virtually nothing about this number, yet it is incredibly significant for numerous reasons. It serves as a special value of a unique function known as the Dirichlet beta function, and it appears in fields such as statistical mechanics, combinatorics, the mass distribution of spiral galaxies, and in a plethora of complex integrals.

As Seán Stewart noted,

“There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan’s constant.”

Two of my personal favorites include:

and

These may seem vastly different from one another, yet both are strikingly simple—though, as we know, appearances can be deceptive. If you aspire to become the next Euler, attempt to find a closed form for this constant! Your name could achieve everlasting recognition.

Upon closer inspection of our list, you may also notice that we are missing the series:

This is known as Apéry’s constant, named after Roger Apéry.

Finding a closed form for that series is an even more renowned problem than that of Catalan’s constant. However, we know more about this number, as it has been proven to be irrational. Even Euler could not solve the challenge of deriving a closed form for this number, suggesting that it is likely quite difficult—perhaps even impossible with our current known constants and methods. The alternating version of ?(3) is equally challenging (in fact, equivalent).

In general, we possess limited understanding of these series when the power is odd. We have yet to consider fifth or seventh powers! As Erdös famously remarked:

“Mathematics is not yet ripe enough for such questions.”

All of the series discussed are part of a larger family known as Dirichlet series, with a specific well-behaved class called Dirichlet L-series.

These infinite series have a robust connection to the distribution of prime numbers because they can all be expressed as infinite products over primes, they exhibit symmetry about a particular line, and they seem to adhere to a principle known as the Riemann Hypothesis.

Each of these L-series or L-functions has its own corresponding Riemann hypothesis, and the simplest of all L-functions is the one Euler began to study in 1734:

When considered as a function of a complex variable, this function is known as the Riemann zeta function, although it has other counterparts with their own Riemann hypotheses. For example, in this article, we examined values of

which is another L-series defined by something called a Dirichlet character of period 4. We have established that ?(1) = ?/4, ?(2) = G, ?(3) = ?³/32. We have no information about ?(4) other than that it is linked to values of a polygamma function, which is equally perplexing. However, it turns out that ?(5) = 5?/1536.

There exists a specific method for extending the domain of these functions so that evaluation at almost all complex numbers makes sense. By doing so, one can demonstrate that ? has zeros at the negative even integers -2, -4, -6, -8,… and that ? has zeros at the negative odd integers: -1, -3, -5, -7,…

This is due to a broader pattern recognized as character parity.

However, both functions appear to have all their other zeros aligned on the same vertical line, specifically at Re(s) = 1/2. The assertion that their non-trivial zeros lie on this line forms the basis of the two Riemann hypotheses attached to them. These hypotheses are merely two from an infinite family of L-functions, all believed to have Riemann hypotheses that maintain their non-trivial zeros along that vertical line in the complex plane.

The entire problem is known as the generalized Riemann hypothesis. If you succeed in solving it, you could win a million dollars, but as someone aptly noted:

“This is probably the hardest way to earn a million dollars!”

The essence of our detour here is that the infinite series we’ve discussed are components of a far larger framework. The mathematicians of India, Leibniz, Euler, and all the other figures in this narrative could not have anticipated this, but now we do. There is a reason why these problems are challenging yet crucial.

And it is not solely about comprehending specific constants. It is about grasping the functions that give rise to these constants.