The Riemann Hypothesis, Demystified

I’ve been obsessed with the Riemann hypothesis for quite some time now and thought about writing an article regarding the same.

The Riemann Hypothesis is one of the 7 millennium problems in mathematics sponsored by The Clay Mathematics Institute of Cambridge. If you prove or disprove it you’ll get accolades and a million dollars. But the most important thing about this is, it serves as a testament to how beautiful the world of numbers is. In this article I’ll try my best to explain it.

Let’s start with prime numbers. What is a prime number? Almost anyone with basic schooling knows this. But here’s the definition again: A prime number is a number whose factors are one and itself. Eg. 2,3,7 etc., are prime numbers and 4, 9 etc., are not.

Let y=f(x)

Now, Let’s define a function f(x) which returns the total number of prime numbers less than or equal to x.

Such that it satisfies the values like,

f(1) = 0

f(2) = 1 (2)

f(3) = 2 (2, 3)

f(8) = 4 (2,3,5, 7)

f(12) = 5 (2, 3, 5, 7, 11)

Although it may seem simple at the first glance, the idea is to generalise the function such that it works for any given x. Be it a trillion, a zillion or a googol.

For simple functions, we can easily define them. For instance, define a f(x) which returns the square of the given number. We can define it as

f(x) = x*x

Now, this works for any given number. But we can’t come up with a simple equation for our prime counting function. The mathematicians over the ages have been obsessed with the distribution of prime numbers and tried coming up with different ways to define this function. That’s exactly what this is about.

This shows the graph of our f(x) which gives the number of primes less than or equal to x.

The graph below shows the other functions which approximately matches the prime counting function. Here the orange line denotes our actual prime counting function π(x). The prime counting function is approximated to x/log(x) which when made some adjustments x/(log(x)-1.08366), becomes more close to π(x).

Source: https://shorturl.at/bsyHM

The above graph might not look smooth but when the numbers becomes larger, it gets smoothened out. From this graph we can notice that prime counting function π(x) is very similar to the function called logarithmic integral function(denoted in violet) li(x). If you ignore those bumps we can see these two functions almost coincide.

Now, Riemann wondered if there is a way to tweak the logarithmic integral function li(x) such that it exactly coincides with the prime counting function π(x). By tweaking I meant plugging in a few constants to the logarithmic integral function just like we modified x/log(x) by adding a constant in the denominator. If we found such a constant or constants, then there is it, we’ve got a function which gives the number of primes less than or equal to a given number. And this would revolutionise the field of mathematics.

Have you seen this weird equation somewhere?

1+2+3+4+……… = -1/12

For those who are coming across this for the first time, it might come as a shock but this is a real thing and it decorates the books covering advanced mathematics which helps us understand the universe.

If you search this on youtube you’ll get all sort of absurd videos proving how this is equal.

Next thing you need to know about is the zeta function.

This function was defined for numbers in which the real part of the number is greater than 1 Re(s)>1. But using a concept called analytic continuity, the domain of the function was extended to Re(s)≤1 as well.

The mathematicians plugged in different values and got a bunch of fascinating results. And if we substitute s with -1, we get this absurd result.

1+2+3+…… = -1/12

How this result came to be deserves a whole article in itself so let’s just leave it here. If you want to know more, you can read about it here — https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF.And this doesn’t make sense as it’s a divergent series but if we dwell deep enough, there are lot of such equations that come up, to which mathematicians just assign a value to.

So now, Let’s see what Riemann did. He started plugging in complex numbers to the zeta function.

Like 1+i, 2+3i etc.,

Where i, 3i represents the imaginary part. When we plug the complex values into the zeta function, we get this beauty.

Now we can see that the value of the zeta function becomes zero for a few values.

We get two types of zeros, one is trivial zeroes and non-trivial zeros. When we plug in -2, -4, -6 etc., (negative even integers) to this function, we get zero and we call them as trivial solutions. Trivial solution refers to something very obvious. One example of trivial solution for 5x+y=0 is x=0, y=0. Non-trivial one here would be (-1, 5) or (1,-5).

Here we don’t care about the trivial solution of the zeta function. And here lies the beauty and this is what the Riemann hypothesis is all about.

According to Wikipedia:

The Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.

And here we’ll ignore all the trivial zeros of the Riemann zeta function which occurs at negative even integers. We’ll only focus only on the non-trivial zeros.

But why, what is the significance of the non-trivial zeros?

Remember when I said the Logarithmic Integral li(x) function looks almost similar to our prime number count function? And how if we plug in some constants to the Logarithmic Integral function, they both converge?

Yeah these values providing non-trivial zeros are those constants which when plugged into the logarithmic integral function gives the prime counting function. The more values we add, the more accurate it gets.

And this is how,

π(x) is our prime number counting function.

li(x) is our logarithmic integral function

π(x) = li(x) — Σ[P(ρ) / ρ] + 1/2

Where, P(s) = 1^(-s) + 2^(-s) + 3^(-s) + 5^(-s) + 7^(-s) + 11^(-s) + …

Basically the prime number counting function is equal to li(x) +1/2 subtracted by summation of P(s) for all values that provide non-trivial zeros in the zeta function. The more values we add the more accurate it gets. And if we add up all values it will be exactly equal to the prime number counting function.

Now, if you prove that any one of the non-trivial zeros lie outside the line where real part is 1/2 then congrats, you have disproved the Riemann’s hypothesis. So far no one was able to prove or disprove it.

Hope I demystified Riemann Hypothesis and hope you had fun reading as well. See you in the next article.

Cheers :)