Leonhard Euler: The Basel Problem

By: Victor Hui

Introduction
Leonhard Euler was a ground-breaking Swiss mathematician and physicist from the 1700's. He made many revolutionary discoveries. However, the one that caught my eye was his solution to the Basel Problem in the year 1734.
The Basel problem was initially posed by an Italian mathematician by the name of Pietro Mengoli in the early 1640's. This problem baffled the even the greatest minds at the time. Branching from mathematical analysis, the Basel problem involved knowledge of convergent series. Convergent series are series where the sum of all terms approaches a limit. As a result of solving a problem that tormented even the brightest mathematicians of the time, Leonhard Euler's and his Solution became exceedingly
popular in the mathematical society at twenty-eight years of age. Over a century later after the solution was discovered and then forgotten, Bernhard Riemann had began to unearth Euler's ideas and bought them to his influential paper in which he defined a single Zeta function in which the Basel problem is also included.

The Problem The Basel problem is defined as the specific summation of all the reciprocals of the squares of the natural numbers. The specific sum of the convergent series is written as such... The Capital Sigma (The large "E") represents the summation of all terms in the series.
The infinite sign represents the series being infinitely long in terms.
And n=1 tells us that the series begins with 1/1 squared and then so on increasing by margins of 1 natural number at a time.
The Basel problem demands for the definite sum of the series in closed form.
At first when we try to add the first eight terms in the series together, we begin to create a number that slowly approaches 1.6.
1/12 + 1/22 + 1/32 +1/42 +1/52 +1/62 +1/72 +1/82
=1 + 0.250 + 0.110 + 0.0625 + 0.040 + 0.027 + 0.020 + 0.015625
= 1.527422052

Leonhard through algebraic genius solved the problem and his sum came up to be: At first, this limit may seem quite confusing What does the summation of all the reciprocals of the squares of the natural numbers have to do with pi?

The Solution
In order for Euler to get this sum, he needed to use a different infinite series. He used the sine function. He then divided both sides by X

By utilizing the Weierstrass factorization theorem, which allows Euler to prove, just like in finite polynomials, the left-hand side of the infinite polynomial is the product of linear factors which are taken from the roots:
We can multiply out this product and then collect all the x2 terms (we are allowed to do so because of Newton's identities), we see that the x2 coefficient of sin(x)/x is But from the original infinite series expansion of sin(x)/x, the coefficient of x2 is −1/(3!) = −1/6. These two coefficients must be equal; thus, Multiplying through both sides of this equation by gives the sum of the reciprocals of the positive square integers.

Visualize
As more terms are added, the value slowly reaches an asymptote that is between 1.6 and 1.7.

Application: Relevance to the Riemann Hypothesis

The Riemann Hypothesis, first published in Riemann's astonishing 1859 paper, was heavily inspired by Leonhard Euler's ideas when he first solved the Basel Problem.
Through the Basel Problem's definition, Bernhard Riemann produced a mathematical notion which is identical to the Basel Problem except for the fact that rather than the reciprocal squared, we have the variable "s".
The Riemann created a function in which currently states that any value of "s" that makes the series equate to zero is either a negative even integer (trivial zeros) or when the horizontal axis equates to 1/2 (called the critical line). The critical line lays in between the horizontal axis equating from 0 to 1. We call this the critical zone. The Riemann Hypothesis asks for a nontrivial zero from the critical zone, a value of "s" that makes the function equate to zero that is not a negative even integer or does not lie on the critical line. This hypothesis was marked as one of the millennium problems. No one has been able to find non trivial zeros off of the critical line and inside of the critical zone. The rules of the millennium problems states that one million dollars is awarded to the mathematician that can find a zero outside of the critical line and inside the critical
zone. The Riemann Zeta Function is graphed on a complex plane.

Conclusion
The Basel Problem was the genesis of Euler's fame. It really defined Euler's mathematical genius and even inspired many more mathematicians past him time such as Bernhard Riemann. Through algebraic genius, Euler solved one of the unsolvable problems of his time. And through his steps we can then learn more about the fundamentals of calculus. When Euler was initially solving the Basel problem, he had came up with using the sine function and slowly equating the sine function with the Basel problem. No one knows how or why he thought of using the sine function but Euler's solution helps us realize that in order to solve major mind-baffling problems, we must have intelligence, inspiration and also creativity. We can apply his solution to help us understand infinite series as well as limits. We can also use his ingenuity to help us solve larger problems in our own time.