Let’s understand π²/6 : The Basel Problem
Objective of this article: Proving that,
Assumed knowledge: Analysis, infinite series, L’Hôpital’s rule
History and Definition
Italian mathematician Pietro Mengoli (known for his work in many infinite series, proving the divergence of the Harmonic series, developing important results that formed the basis for Newton and Leibniz, etc.) posed a problem in 1644. The problem asked to find the value of,
Many great mathematicians tried to find it’s solution. In 1655, Wallis thought he knew the solution till 3 decimal places, in 1689 Jakob Bernoulli claimed that the solution must be less than 2, Johann and Daniel Bernoulli said the answer was about 8/5 in 1721, and around the same time Goldbach claimed that it is definitely between 1.64 and 1.666…
The difficulty in solving this series was that it converges very slowly, making it hard for mathematicians then to figure out the accurate solution.
It was in 1735 that Leonhard Euler posed a complete solution to this problem stating that the sum is π²/6. As many great minds struggled with this problem, this brought Euler a lot of immediate fame among the mathematicians. Having been born in Basel, this problem was then on called ‘The Basel problem’ or ‘The Basel theorem’.
Euler then published a more rigorous proof in 1741 and then went to publish a third proof in 1755. His proofs then inspired Weierstrass and Riemann to develop analysis and the zeta function, defined as,
Re(s) > 1
This then led to the discovery of the Riemann hypothesis, which is still one of the most important unsolved problems in mathematics.
Now, with enough historical knowledge, let’s move on to the proof of this problem.
We will go through two of many proofs that exist, one of which is Euler’s. So, let’s begin with the first one.
Proof 1
First we will try writing sin x as an infinite product of linear factors.
Differentiating with respect to y,
Substituting y = -ix or y² = -x²
Using the Euler’s Formula,
Substituting this back to the previous equation,
Now, if we take x = 0, the LHS would be the Basel problem. But the RHS will become 0/0. So, to solve this let’s use L’Hôpital’s rule.
Q.E.D.
Proof 2
This was the first ever proof proposed by Euler in 1735 that brought his fame. This is, honestly, the easiest proofs among many that I had come across.
So, once again, we’ll write sin x as an infinite product of linear factors.
Also, writing the Maclaurin series for sin x,
Comparing the coefficients of x³ from the two series of sin(πx) written above,
Thus,
Q.E.D.
[Note: Why can we “factor” sin x into linear terms? This can be understood by depending on Complex Analysis, specifically by the concept of Weierstrass Factorization Theorem.]
Other proofs
While researching this topic, I came across a million different proofs from a variety of areas of mathematics: Complex Analysis, Calculus, Probability, Hilbert space theory, etc. One included the use of integrals and calculus, (like the one by Tom Apostol in 1983 : https://fermatslibrary.com/s/a-proof-that-euler-missed), one used Fourier Analysis and another, Parseval’s Theorem.
Euler’s solution to the Basel problem had a profound impact on the development of mathematics. It established a deep connection between number theory, complex analysis, and algebraic geometry. In addition to its mathematical significance, the Basel problem has also captured the imagination of people outside of mathematics. It has been the subject of many popular books and articles, as well as numerous television shows and movies. It has become a symbol of the beauty and power of mathematics.
