Millennium Problem: Riemann Hypothesis

A prime number is a number that is divisible only by itself and 1. Some examples being 2, 3, 5, 7, 9, 11, 13.…  The great mathematician Euclid proved centuries ago that there are an infinite number of them continuing without pattern ad infinitum. However, their real importance is given by what is known as the Fundamental Theorem of Arithmetic. It states that every number can be uniquely expressed as a product of prime numbers. Let’s take the example of 364 = 2 × 2 × 7 × 13. This is the only way that 364 can be obtained by prime numbers! Therefore, they are the atoms or building blocks of numbers.

But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function.

The Riemann zeta function takes inputs not only real numbers but also  complex numbers — meaning they have both “real” and “imaginary” components — and yields other numbers as outputs.

In fact, the Riemann zeta function is convergent for all values when real part of s is greater than 1.

For example, Euler famously showed that

The important point being that besides the singularity at s =1, we are able to find values everywhere for the Riemann zeta function even when it is divergent because it is a Holomorphic function and hence amenable to analytic continuation which expands the domain of the function. 

It is to be noted that there are many trivial solutions for the Riemann zeta function. For example, it has been observed that for all negative even numbers s, ζ(s) = 0.The Riemann hypothesis asserts that all non-trivial Zeros of the equation

 ζ(s) = 0 lie on the vertical line where the real part of s is ½.

This has been checked for the first 10,000,000,000,000 zeros.

Riemann discovered a formula for calculating the number of primes up to any given cutoff by summing over a sequence of these zeros. The formula also gave a way of measuring the fluctuations of the primes around their typical spacing — how much larger or smaller a given prime was when compared with what might be expected.
However, Riemann knew that his formula would be valid only if the zeros of the zeta function satisfied a certain property: Their real parts all had to equal ½. Otherwise the formula made no sense. Riemann calculated the first few nontrivial zeros of the zeta function and confirmed that their real parts were equal to ½. The calculation supported his hypothesis that all zeros had this property, and thus that the spacing of all prime numbers followed from his function. But he noted that “without doubt it would be desirable to have a rigorous proof of this proposition.”

A proof that it is true for every non-trivial zero would shed light on many of the mysteries surrounding the distribution of prime numbers.
I would like to thank Prof Keith Conrad for his many insightful comments which made the article better and technically more accurate.
Any remaining inaccuracies are entirely my responsibility.