First identity – Sum to infinity.

The first identity shows the convergence of the Möbius function. It seems that as we form a sequence of infinite terms, they cancel out each other to give the sum of zero.

Second identity – Sum to infinity with a logarithmic function.

Another amazing identity. Again it converges to negative 1 but this time it involves a logarithmic function.

Third identity – The occurrence of pi.

Each identity simply fascinates me even more. This time, the irrational number pi appears as the denominator of the quotient that the series converges to. I certainly don’t see any link between the Möbius function and circles.

Fourth identity – The product series an Eulerian number.

The Möbius function ends with a bang! I am shell shocked at how it connects so many aspects of mathematics in an elegant identity. I need not say more.

We wrap up our exploration of the Möbius function with it’s involvement with Palindromes. A palindromes are numbers that read the same left to right and right to left like, 12421. This discovery, credited to Jason Earl from Texas, shows what we get when we apply the Möbius function to certain Palindromes. One such example is to the number 15,891,919,851 and each right truncation of its digits.

For a palindrome such as this, we call it a möbidrome. Rest assured, this is not the only one. In the world or integers out there, mathematicians are consistently looking for möbidromes lurking in the company of palindromes.

And so our journey with the Möbius function ends here. The innocent action of putting positive integers into hats brought forth lovely identities which are suggestive of its uses in other fields of mathematics. May I humbly mention again the mathematician August Ferdinand Möbius who created a function, so intriguing yet fascinating, chaotic yet elegant.