THE RIEMANN HYPOTHESIS REMAIN UNSOLVED IN TODAY'S WORLD

 

THE RIEMANN HYPOTHESIS REMAIN UNSOLVED IN TODAY'S WORLD 

Not many mathematical problems remain unsolved in today's world, yet there is one problem that has captured the attention of mathematicians for decades, if not centuries. It is known as The Riemann Hypothesis. A solution to this single hypothesis would prove hundreds of other theorems.

As the mathematician Peter Sarnak said :

“If [the Riemann Hypothesis is] not true, then the world is a very different place. The whole structure of integers and prime numbers would be very different from what we could imagine. In a way, it would be more interesting if it were false, but it would be a disaster because we've built so much around assuming its truth.”

The Riemann hypothesis is a mathematical conjecture that was first proffered by German mathematician Bernhard Riemann in 1859. The Riemann zeta function was first encountered in the study of Complex Analysis and is important due to its remarkable connection to other fields in mathematics, such as number theory and matrix theory.

The Riemann Zeta function is a function of complex numbers, and its graph is plotted in a complex plane, also known as the Argand Diagram. In the graph, each point represents a complex number where the horizontal axis represents the real part (Re) and the vertical axis represents the imaginary part (Im).

The Riemann Zeta Function is defined as,

ζ(s) = Σ( 1 → ∞ ) ¹⁄ₙˢ

Where s is a complex variable, s = a+ib.

(a is the real part; Re(s) = a

and ib is the imaginary part; Im(s) = ib)

[This equation only applies when Re(s) > 1, for other values of Re(s) we need to use analytical continuation. Those equations have been avoided due to their complexity]

To calculate the zeta function for complex values of s, special numerical algorithms and techniques are employed. There are various computational methods to compute the zeta function, and these methods are implemented in mathematical software libraries like Mathematica, MATLAB, or Python's SciPy.

(To avoid calculational complexity, we have just given an idea about the properties of this function based on its complex domain range without including specific reasoning.)

When,

Re(s) > 1, the value of the function is finite, meaning it converges to a certain value.

Re(s) = 1, the function is undefined.

Re(s) < 0, We can obtain some values by using analytical continuation. (Not necessary for this article.)

If s = σ = -2n, where n is a positive integer, the function results in zero. These zeros are called trivial zeros.

So, we can say that ζ(s) doesn’t give any zero when Re(s) > 1 or Re(s) = 1 even when Re(s) < 0 ≠ -2n and the trivial zeros only come when s = −2, −4, −6 ⋯ . Therefore, the rest of the zeros (if exist) lie on a single strip, known as the Critical Strip, where Re(s) is in between 0 and 1 (Only possible case available). These are known as non-trivial zeros. Their location is connected with the distribution of prime numbers. Riemann calculated some of these zeroes in his 1859 paper, and he found that in all those non-trivial cases, Re(s) = ½ or, if you want to think of it in terms of our graph coordinates, they all lay on the same vertical line.

Now there comes the main problem:

“Do all the non-trivial zeros exist on the ½ line?”

That is the famous “Riemann Hypothesis” which has become a holy grail for mathematicians. It has been over 163 years, and not even a single mathematician could solve it to this day. It’s been titled “The most famous problem in all of mathematics” and if one can solve it, he will be awarded a million dollars. A solution to this problem will stiffen humanity’s understanding of the most fundamental objects of mathematics and uncover the secrets behind the distribution of prime numbers. 

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