Euler Brilliance: From the Basel problem to the Zeta Function

Image source https://www.historyhit.com/leonhard-euler-one-of-the-greatest-mathematicians-in-history/

Leonhard Euler, a mathematicians being considered the GOAT, with 800 papers during his life time, giving us a treasure trove of knowledge in many fields. Beside his famous Euler identity, one considered the most elegant equation in the world :

One equation I really love is the Euler product , it is elegant as it create a bridge from the analytic function with primes, the foundational numbers of number theory, yet the most mysterious one.

In this blog we will go to his story, what did he think and what is his intuition and the background stories behind this stunning equation.

Fermat prime producing function

It also started from the French mathematician Fermat who is famous for his quote for Fermat last theorem:

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain

In one of his paper he claimed that all of the number in this form is prime :

Yet this is true for N=0 , 1, 2,3,4. However, Euler soon realized that for N=5, this failed as it has a factor 641 , a ridiculous mental calculation !

Yet the idea of prime producing function give him a motivation , he asked himself

Why don’t I make one ?

With this in mind he actually came up with the polynomial

But he soon realize this only hold until n = 39 , with n= 40 it is not a prime number. However, the idea of prime generating function illuminated him for the famous Euler Product, which we will discuss later at the end of the blog.

You might think the quadratic equation here is just a coincidence, actually, it lies on a deeper Math, connecting to the straight line in the Ulam Spiral !

The idea came from Basel theorem

At that time , a famous problem , easy to understand but never been solved

At that time, there are no Fourier transform or Parseval to solve this Basel equation. Yet, Euler has the very unique and brilliant solution :

Firstly, he took a look at the sine function he knew two facts

where the first on is derived by the Taylor series of sine function

He though, a brilliant yet not standard way to continue from this, what if we treat sin(x) as the polynomial, with …. infinite degree . If we do so we will have

As the solution of the sine function is symmetry with the y axis, we obtain

However, when we take the coefficient of any degree here it is infinity, despite C might be small enough . Euler bypass this by write as this, where C’ is any constant

Taking the coefficient of x² of LHS, we have

This is derived from Taylor series

For RHS, we expand and obtain :

Taking the coefficient of x²

Idea of Zeta function and Euler product

After disporve Fermat claim about prime generator , Euler wonder himself of whether he can find one. He ontained that polynomials is not enough to create a real prime generator.

He think that the prime must be very close related to natural number, he quickly write on the paper the first line

He realized this is not converge so he think that may be we can take the sum of reciprocal instead, with the idea of Basel problem, he think , may be instead of the power 2 in the denominator, he can raise to the power n.

That’s exactly what he did , he used the greek letter, zeta to denote

Now he played with this function starting with number two — the first prime :

He compared this two function, quickly , he subtract the original function with the new one

Here, he obtained that all of the number with denominator which base is the multipler of 2 is eliminated. He repeat one more time

Repeating the process, we will have

Which equivalent , QED .

Euler shows us that the zeta function has deep relationship with prime numbers, a number that is very hard to predict making it the most unexpected number in number theory. Thus making the study of zeta function becomes very important.

As you might know the zeta function nowadays, are usually not this type but the Riemann Zeta function, instead of the natural power, but extending to the complex plane.

The Riemann Hypothesis is considered holy grail of number theory, which correctly predicted the prime using the counting prime function. However, this is still remained as the open problem ( 11 April 2026)

I will create a blog about Riemann Zeta function in the near future!

Conclusion

Mathematics comes from the most unexpected thing. From the reckless claim of Fermat, to the idea of zeta function from Basel, creating Euler product, expanding to complex plane which leads to the holy grail — Riemann Hypothesis, each step is pure intuition and creative.

Thus , from this blog , we can deduce that to keep the open mind and be creative , think outside of the box is integral not only in Math or Science but life in general.

Looking back, what if they scared of being critisize and choose safer path ,

What if Fermat does not make such claim ?

and

What if Euler does not apply polynomials expansion for sine function ?