Riemann Hypothesis

Ryn had said this after thinking very carefully. So the professor's anger calmed down a little. He said in a slightly higher tone than Ryn,

--"Come inside. Don't even try to delay further."

Ryn stepped inside the classroom. But then everyone's attention fell on him. Well, Ryn was quite good looking but his average look was quite boring. That's why the girls avert their eyes from him. Now no one was looking at him.

Feeling this makes Ryn feel a little comfortable. And he goes and sits on the back seat. In front was Math teacher William Sir, whose age would be 49 years. He was carefully opening his book. And then he looked around and saw that everyone was engrossed in their conversation.

Seeing this, Willian sir became very angry. He let out a scream which echoed throughout the class. That too more than 4 times.

--"Shut up. We have come here for study, not to gossip. All worthless students I get this year."

Everyone was thinking that perhaps William Sir would become silent now. But maybe he didn't like Ryn coming late and then gossiping about everyone. That's why he continued looking towards everyone till now.

But then his eyes fell on Ryn. Who was picking up his pen because it had fallen down. Don't know why but William Sir got angry at him. Pointing straight back he said,

--"Oh hello! The boy from the last bench. Come forward."

Ryn first looked around and then towards the front. When he saw that William Sir was looking at him, he stood up. Everyone was just busy laughing at Ryn's look. Just then Ryn asked,

 --"Me?"

 --"Yes, you. Come quickly."

Ryn was feeling nervous. But he gathered all his courage and went towards the front. Totally 304 students were sitting in the class. When he went to the front and saw the entire class, Ryan's breath stopped. So many students!

Just then William sir gave Ryn a black marker and said very harshly,

--"Tell me about the Riemann Hypothesis. Imagine that you are taking a class right now. In which you have to explain to your students the basic knowledge you have about the Riemann Hypothesis. Now you start writing. Whether you write or explain, it is only your choice. Depends on you."

William Sir had already thought that no matter what happens, no one can explain even the slightest glimpse about the Riemann Hypothesis, let alone its basics. Riemann Hypothesis is a difficult and very confusing math.

This is becoming more and more difficult every year. Where students start running away after seeing this. That's what Ryn had to do today. Suddenly everyone became silent, starting with William Sir. Because every student knows that Riemann Hypothesis? It is very difficult to solve this.

This is harder to do than it looks. And how can those people do this? Because those people had only heard a little about Riemann Hypothesis. The remaining question is that no one in the class knows about it.

If seen, people learn a lot from this in the 5th year and that too a little. The classroom which was quiet now suddenly started echoing with words of laughter. Because everyone thinks that Ryn does not know anything about the Riemann Hypothesis.

Maybe now William sir will also teach him a good lesson. Everyone started thinking the same thing. William sir also knows that Riemann Hypothesis was not in Ryn's control. He would be Genius only if he solved the Riemann Hypothesis in his first year of college.

But looking at everyone, Ryn's heart was also feeling nervous. But he still gathered courage. And then he started thinking about something while staring at the ground for a few seconds. When William sir noticed Ryn's actions, he was about to move forward to say something rude and hurting.

Then suddenly Ryn looked up. Adjusting his glasses he took a deep breath. And then he began. There was a confidence visible in his words. A different spirit! Looking at Ryn now no one can tell that he is an introvert person.

Ryn started speaking with a very attracted way,

--"In mathematics, 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

Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named."

He said this in one breath. Suddenly everyone started listening to him very attentively. Because in any case, some students had some knowledge about the Riemann hypothesis. But no one knows that a person who looks like Ryan also has knowledge about mathematics like Riemann hypothesis.

Only then Ryn started speaking further. William sir was also a little worried. Suddenly the entire atmosphere of the class became quiet, and everyone started listening to Ryn standing in front.

--"In this work we are studying the properties of modified zeta functions. Riemann's zeta function is defined by the,Dirichlet's distribution

ς(s) =

∞

n=1

1

ns , s = σ + it (1)

absolutely and uniformly converging in any finite region of the complex s-plane, for which σ ≥ 1 + , > 0. If σ > 1

the function is represented by the following Euler product formula

ς(s) =

p

1 − 1

ps

−1

(2)

where p is all prime numbers. ς(s) was firstly introduced by Euler 1737 in [1], who decomposed it to the Euler

product formula (2). Dirichlet and Chebyshev, studying the law of prime numbers distribution, had considered this in

[2] . However, the most profound properties of the function ς(z) had only been discovered later, when the function

had been considered as a function of a complex variable. In 1876 Riemann was the first who in [3] that :

ς(s) allows analytical continuation on the whole z-plane in the following form

π−s/2

Γ(s/2)ς(s) = 1/(s(s − 1)) +

+∞

1

(xs/2 − 1 + x(1−s)/2 − 1)θ(x)dx. (3)

where Γ(z)- gamma function,

θ(x) = ∞

n=1 exp(−πn2 x).

ς(s) is a regular function for all values of s, except s=1, where it has a simple pole with a deduction equal to 1,

and satisfies the following functional equation

π−s/2

Γ(s/2)ς(s) = π−(1−s)/2

Γ((1 − s)/2)ς(1 − s) (4)!"

William Sir's eyes were full of surprise. Because Ryn knows everything, not just the basics, but even more details! He really turned out to be Genius !

--"This equation is called the Riemann's functional equation.

The Riemann's zeta function is the most important subject of study and has a plenty of interesting generalizations.

The role of zeta functions in the Number Theory is very significant, and is connected to various fundamental

functions in the Number Theory as Mobius function, Liouville function, the function of quantity of number divisors,

the function of quantity of prime number divisors. The detailed theory of zeta functions is showed in [4]. The zeta

function spreads to various disciplines and now the function is mostly applied in quantum statistical mechanics and

quantum theory of pole[5-7]. Riemann's zeta function is often introduced in the formulas of quantum statistics. A

well-known example is the Stefan-Boltzman law of a black body's radiation. The given aspects of the zeta function

reveal global necessity of its further investigation.

We are interested in the analytical properties of the following generalizations of zeta functions. It's a very basic rules and terms unfortunately. But for like us, students won't get any benefit for knowing this some simple math or rule of Riemann Hypothesis. But in our country, it's enough for 1st year students. "