Riemann Hypothesis

Now I am definitely not an expert in this field, and in fact even the experts aren't really experts in the conventional sense. No one is an expert on it in the conventional sense, it is still unsolved. Over 150 years old and it still remains unsolved not for the lack of trying! In fact it is so important to mathematicians that the Clay Mathematics Institute has put a $1,000,000 bounty on its head (that is, you get $1,000,000 if you manage to solve it).


But what actually is the Riemann Hypothesis? It is a conjecture about the location of the non-trivial zeros of the Riemann Zeta function, it states that all the zeros should lie on the critical strip 0.5+it. "Oh yeah!", I hear you cry, now you get it, obviously. I will explain what this means properly later on in this post. But first I will state what it means. If true it implies a lot of things about the distribution of prime numbers, and as you may or may not know they are very irregular and very difficult to find as the numbers get very, very big.


To track back to my earlier point, what is the Riemann Zeta Function ( it is denoted as ζ(s), ζ being the Greek lower case from which z was derived)?

Where s is an imaginary number, a+ib.

This requires that you understand sequences and series, imaginary numbers and imaginary exponents. The real intrigue of this comes from the fact that it can be represented by Euler's product.

As you may or may not notice this is comprised of the prime numbers, this means that there is a sort of subliminal link between the natural numbers and the prime numbers. This showed that the prime numbers were not just positioned randomly and are not merely the building blocks to numbers but there is an actual link between them and the natural numbers.


The Riemann Zeta Function on its face doesn't look too difficult, I mean it is just an infinite sequence, even with a complex power you'd expect this to be possible and even pretty easy. But that is not the case at all, part of the reason is how sporadic complex exponents can be, and although it is not too difficult to find solutions (using a high powered computer thousands can be found each hour) it is incredibly, incredibly hard to find a proof for all the solutions.

The plot of the Riemann Zeta Function, the red line is the
real part, the blue part is the imaginary part.

You can see, this function seems to have little to no consistency to it, but a fair amount is known about the function. A lot of the zeros do actually satisfy the hypothesis, over 10 trillion of them in fact. And you'd think that is a proof alone, but as it often involves an iterated log (a log of a log, log(log(x)) and this increases very, very, very slowly in fact log(log(10,000,000,000)) = 1, so 10 trillion really isn't anything. If it is still holding true for log(log(x))>40 there may be a greater unanimous opinion on the truth of the hypothesis.


Every mathematician worth his salt has had an encounter with the Riemann Hypothesis and it has withheld every single attempt thus far. The maths used to try and tackle the problem is so complex that entirely new branches of mathematics have been created to deal with it, this maths to laymen has literally nothing, at all, to do with the prime numbers. It is so complex and far away from the problem that it almost boggles mathematicians minds, but it consumes them, it is their passion and life.


Prime numbers are the passion for many and the Riemann Hypothesis is merely an extension of that, and hopefully it will be solved in my life time.


If you have caught the prime number bug I suggest you read the excellent book by Karl Sabbagh called Dr Riemann's Zeros.