The Riemann Hypothesis Explained

by Harry Leggett

I have been studying maths for years and whenever I come across a question I cant complete I often come to the same conclusion… This must be the hardest maths questions in the world. However sadly within 30 seconds of someone explaining to me that I have been doing the question entirely wrong I realise that it is entirely possible just the lack of my mathematical ability was my limiting factor. Having studied maths at A level and further into a second a level I have realised the broad span of lives that maths touches. I understand the mindset that maths is useless and “When will I ever need this” is a common question to those who are not so excited by the subject. However in my studies I have learnt all about many previous mathematicians and current mathematicians who battled with questions not only for minutes, or hours or days but years. This got me thinking about what kind of questions they would do, are they just questions out of a textbook, and who makes up questions which they cant solve! With this I did some research and quickly came across something called the Millennium prize problems. In 2000 seven questions were set by the Clay Mathematics Institute, and the incentive was that if a correct answer was provided by anyone, then a prize of $1 million would be given. 17 years and 11 months on… only one has been solved. Now if you have not come across these problems before then I would highly recommend looking them up however I will attempt to give a brief and basic (to an extend) evaluation of my favourite one.

Riemann Hypothesis

Before I start this I would like to write a few simple questions.

1.) 1+2 =

2.) 1+2+3=

3.) 1+2+3+4+

And so on…

What would the answer to this be if I continued on to infinity.

The Riemann Zeta function, assigns a certain number of any value of s and is shown by 1/1^s + 1/2^s + 1/3^s + … So if we make s = 2 we get 1 + 1/4 + 1/9 + 1/16. This is a convergent series, meaning if you take the sum of the first n terms it will converge to a specific number. This is called the limit of the series. However if we were to keep the series going until infinity it was shown by Euler that it equals pi^2/6. If we make s = 3 you will get 1 + 1/8 + 1/27 + 1/64 and so on. If we are to continue this on until infinite this has been proven to converge towards 1.2020569031… This is known as Apery’s constant. We can do this with all kinds of numbers, getting and bigger and bigger, or we could take negative numbers. This interesting thing is that if we use -1, you can use basic calculus indices and so you take the inverse of an inverse so it becomes positive, so the series you then get is 1 + 2 + 3 + 4… 

Now this is not an uncommon series. This is a famous series as not only is what we use when learning to count from a very young age, it is also fascinating in the highest level of mathematics. It is the sum of the all natural numbers question. It is a divergent series, we can conclude this as the difference between the terms is increasing and also proof by proving its not convergent. However we can assign a value for when s=-1 and this is what Riemann said in his initial paper. Reimann worked out that if you make s any value greater than 1, all of the series are convergent. Reimann worked out that you can extend all values for any real number, except 1. Reimann couldn’t find a value for when s=1. This is called a singularity. When s=-1 you get a value of -1/12! This is a very illogical answer, I believe for two reasons. Firstly how can adding positive numbers together equal a negative, and secondly how can adding whole numbers equal a fraction! So Reimanns questions (literally the one million dollar question), for what value of s = 0? It has been calculated that all negative even numbers equal zero, however there are more. We know that the other values are within the “critical strip”. The critical strip is defined as “The region , where is defined as the real part of a complex number . All nontrivial zeros (i.e., those not at negative even integers) of the Riemann zeta function lie inside this strip.” Within the critical strip there is another strip which lies through the middle which goes through the point 1/2. It has been theoretically concluded that all the zero values are within the strip, however it has not been concluded that all zero points are on the middle line. All zeros that have been found excluding negative even numbers have been on the line.