Ode to Euler

Source: USNA

Leonhard Euler was the most distinguished and popular 18th century mathematician. His contributions to Mathematics are as profound as they are diverse- he worked on number theory, algebra, analysis, geometry, statistics, linear algebra, applied mathematics, and so many other subjects within the discipline. Just to give you an idea of how prolific Euler was: He started writing for the journal of St. Petersburg academy from 1727, and continued doing so till he died in 1783; however, the journal continued to publish Euler’s articles for 48 years after his death!

Euler is too tall a giant for me, or for most other math enthusiasts to stand on the shoulders of. Of him the great French scholar and mathematician Pierre-Simon Laplace said: “Read Euler, read Euler, he is the master of us all”.

Like my last article, this one too is inspired by a book I am reading currently. It’s called “Euler: The Master of Us All” written by Professor William Dunham. I recommend everybody reading this tiny article read the book too, because the book contains many formulas and theorems Euler derived, although even these “many” are a small proportion of Euler’s achievements.

Euler ‘s biggest achievements are in analytic number theory and infinite series. Let me start with the definition of a the most well-known function in the subject, known as the Riemann zeta (or Euler-Riemann zeta) function:

Where ‘s’ is any complex variable (if imaginary part is zero it is real) . Euler was interestingly the first one to study such types of functions (for even positive integers). His association with it started with the famous “Basel’s problem”, where one was asked to find the sum of the function when s=2. This problem had challenged the Bernoulli's even. But alas, Euler found out the following result, amazing and yet completely non-intuitive:

Euler’s first proof for this was a totally non-rigorous one and at that time was not completely accepted by mathematicians, although he did receive a lot of acclaim. I’ll explain the non-rigorous proof because it is easier than Euler’s rigorous proofs to the result. To prove this, let us introduce two infinite expansions for sine of an angle:-

This is the Maclaurin series expansion (or Taylor expansion at x=0) for sine of x

This was only proved many years after Euler used it for Basel’s problem

Now using the two Euler did something amazing. He first expanded the second infinite product and equated it to the first divided by x:

Right-hand side can be obtained by expanding the terms of the infinite product for sin(x)/x above

By equating the coefficients for x² from the two sides of this equation he got:

And from this Leonhard Euler concluded in dramatic fashion-

Similarly Euler went ahead and proved results for the function for 4,6,8,etc.

Also known as the Stefan-Boltzmann Law or Wien approximation in Physics

Another amazing result!

This was later generalized for even positive integers using Bernoulli numbers-

For all positive integers or natural ‘n’ and where Bernoulli numbers are explicitly defined as follows:-

First proved by Louis Saalschütz in the year 1893

You can also try out some other really cool stuff Euler proved in his papers, like an expression for the Zeta function using primes(which is divergent for s=1, first shown by Jacob Bernoulli) and the famous Wallis product for pi:

If you find this hard, Don’t worry, this is something beyond my scope of understanding too!

Wow, John Wallis, you genius!

Now let us try another of Euler’s adventurous proofs- one in the field of Number Theory, something , much easier, that related perfect numbers to Mersenne primes, known as the Euclid-Euler theorem.

Euclid has proved in his proved the following result (Euclid, Prop.IX.36):-

And from this Alhazen conjectured that every even perfect number is of this form. It was only Leonhard Euler who proved this conjecture and in his proof he used the divisor function for positive integers which is defined as follows:

The sigma or divisor function (d|n stands for d divides the number n)

Suppose N is even and perfect. Factor out all powers of 2 to write N as follows:

Since N is even k > 1. Also because N is a perfect positive integer we know:

At the same time since 2^k-1 & b are co-prime, using a property of co-primes:

Now using the previous two equations for the divisor function of N we get:

The left-hand side is in its simplest form (or in lowest terms) but the right-hand side is not yet clear, so we can write that for some c ≥1 we have:

Now what Euler decided to do was to consider the two cases for c(> or =1):

c > 1

Each of the whole numbers 1,b,c,-1+2^k divides b. Now considering pairwise equalities for these numbers we get:

1. 1 ≠ b for otherwise N is 2^(k-1) which is not possible because no power of 2 can be perfect.
2. 1 ≠ c for we have stipulated c to be c > 1.
3. 1 ≠ (2^k)-1 for otherwise N = b which makes N odd but N is even.
4. b ≠ c for if these were equal then 1 = (2^k)-1 which is not true.
5. b ≠ (2^k)-1 for otherwise b=cb which implies c=1.
6. c ≠ (2^k)-1 for otherwise b=c² but the sum of divisors of b is c² + c and in this case the sum is more than or equal to c² + c +1 which is absurd.

Therefore the numbers 1,b,c,(2^k)-1 are four different divisors of b. Thus:

Therefore this case is entirely false since this is a contradiction. Then considering c=1-

c = 1

This implies the only divisors of b are 1 and b, or in other words, b is prime.

Thus we have shown that if N is an even perfect number, then N = (2^(k-1))b

⇒ N=(2^(k-1))*((2^k)-1) where (2^k)-1 is prime. The necessity of Euclid’s condition is thus established and hence Euler proved it.

Euler did so many things in every field of Mathematics that there is no space big enough to fill in all of his achievement. I suggest everybody reading my article to not only read Prof. Dunham’s book but also try catching hold of a translated version of Introductio in Analysin Infinitorium and also Opera Omnia which according to me summarize who Euler really was- an adventurous scientist and carefree explorer. He gave non-rigorous proofs to so many infinite expansions without the use of Calculus and used those in his papers on Calculus thus independently proving a lot of great Math theorems.

I really wish somebody gets inspired by this article like I did and reads his work. Truly, Euler is the “Master of us all”!

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