About things that don't add up...or do they?

Recently I interviewed an Irish data scientist for a new position in my AI/Advanced Analytics team. What caught my eye in his CV where (1) his master thesis on the Riemann Zeta Function and (2) his experience as a bar tender in an Irish pub. The latter subconsciously tipped the scale to make him an offer as it fondly reminded me of a post graduation van tour with schoolmates around Ireland in 1988 where we had our daily visits to Irish pubs with plenty of Guinness & cheap Paddy's whiskey.

I always enjoyed reading bios about mathematicians, numbers and math in general. Riemann is linked to so many interesting stories that I came across over time, not unlike the rabbit hole you go down when you start watching cute cat videos on YouTube and their recommender algorithms gets you to the more bizarre links like the Netflix docuseries "Don't F**k with Cats: Hunting an Internet Killer" (it's worth watching though).

Back to the bartender's master thesis - the Riemann Zeta function is linked to the Millennium Prize Problems by the Clay Mathematics Institute that offers US$1 million each for anybody who can prove one of the seven unsolved mathematical problems: 1. the Birch and Swinnerton-Dyer conjecture, 2. Hodge conjecture, 3. Navier–Stokes existence and smoothness, 4. P versus NP problem, 5. Poincaré conjecture, 6. Riemann hypothesis, and 7. Yang–Mills existence and mass gap. 

To date only the Poincaré conjecture has been solved by Russian Grigori Perelmann, who famously declined the award and prize money as he felt he was just one contributor to solve this problem.

So, what is the Riemann hypothesis? It centers around the Riemann zeta function ζ(s), defined as below.

This zeta function was first defined by German mathematician Riemann in 1859, and build on a similar function by Swiss mathematician Leonard Euler who analyzed the function for real numbers of s. Euler in particular proved the so-called Basel Problem, stated by Italian Pietro Mengoli in 1650, which asked for the precise sum and proof of the function for s=2. Euler proved that the result is surprisingly simple:

This result doesn't look like rocket science now but back then it was, sort of, as it's a fraction of π, i.e. a number usually associated with circles.

Riemann extended the Euler zeta function from real to complex numbers, i.e. s may be any complex number (s = a + b i, with a, b real numbers and i2 = -1 being the imaginary unit) . This function has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of -2, -4, -6, -8, .... - these are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every non-trivial zero of the Riemann zeta function is 0.5. I.e. all non-trivial zeros lie on the so-called critical line as shown in above graph.

Prove this hypothesis and one million dollars is yours. To note that the Riemann hypothesis is also one of the famous list of 23 unsolved problem German mathematician David Hilbert presented in 1900 of which only 8 have been solved to date.

The Riemann hypothesis has important implications e.g. for the distribution of prime numbers. And in extension for cryptography. And many other applications/examples as you can find in this link, none of which will makes much sense to the non-mathematician.

Brute force methods have been applied to find a counterexample to disprove the Riemann hypotheses, i.e. using powerful computers to find a non-trivial zero not on the critical line. So far none has been found, although it has been computed that the first 10 trillion non-trivial zeros lie indeed on the critical line.

Whilst the Riemann hypothesis is a touch too academic for a LinkedIn post, what makes it more mainstream is the special case s =-1 for the Zeta function for real values, i.e. the simple infinite sum of natural numbers 1+2+3+4+5+...

Anecdote has it that German mathematician Carl Friedrich Gauss was quick witted when his school teacher asked his class for the sum of 1 to 100 and lil' Carl immediately replied 5,050. Instead of adding the 100 numbers up, he regrouped them into 50 equal terms of 101: (1+100) + (2+99) +(3+98) + ... + (49+51) = 50 *101 = 5,050. In general terms, this means that any partial finite sum 1+2+3+...+n equals n (n+1)/2.

If you wonder what the Zeta function adds up to when you sum to infinity, common sense would say infinity. But to confuse the common man, this series can indeed be added up in a way that yields the following, rather stunning, result:

The Youtube math channel Numberphiles has an interesting "popular" proof that went viral (sort of) with over 8m views:

In short, their reasoning goes like this:

If you doubt the math skills of these YouTubers you might be more convinced if you learn that India's most famous mathematician Srinivasa Ramanujan came to the same conclusion in a famous letter to English mathematician G.H. Hardy in 1913. Conscious of the seemingly absurd result he wrote to Hardy "If I tell you this you will at once point out to me the lunatic asylum as my goal."

The major fine-print in the above is that the equal sign in ζ(-1) = -1/12 is not the same as you were taught in primary school. This equation only holds true when special summation methods are used that assign finite values to divergent series. "Ramanujan summation" and "Zeta function regularization" are such methods. E.g. in Numberphiles' "proof" above, the function S? is alternating between 0 and 1, depending on when you stop summing up at even or odd numbers. But it doesn't converge to 1/2 and is divergent. Equally is ζ(-1) divergent. So, it's a bit of a neat "party-trick" with some smoke and mirrors, but entertaining and attention-catching none the less.

If this leaves you confused given that we'd usually expect mathematics to be rather rigorous and not allowing such seemingly illogical methods, think again...as mathematicians have also introduced the equally bizarre imaginary number i, with i2 = -1.

I get why this concepts make senses in mathematics but don't have the brains nor time to research this in more detail. And I really love Ramanujan's life story (read the book or watch the movie "The man who knew infinity"), that I like to belief that it's genius.

I will leave it to the bartender to write a follow-up LinkedIn article to explain how these special summation methods do work ... .