Volume 2 Academic Career Chapter 768: The solution to the Riemann hypothesis!
In his research on the BSD conjecture, Chen Zhou actually did not think that he would get the inspiration to solve the Riemann hypothesis.
Moreover, in Chen Zhou's original plan, after solving the BSD conjecture, he would turn to the study of the grand unified theory of physics and completely end this topic.
But plans can never keep up with changes.
Chen Zhou couldn't just put aside this exciting flash of inspiration and ignore it, right?
That is obviously impossible.
After getting the inspiration for solving the Riemann hypothesis, Chen Zhou decisively continued his research on the mathematical topic, putting solving the Riemann hypothesis at the top of his agenda.
Even the research paper on the BSD conjecture that Chen Zhou planned to complete was put on the back burner.
The Riemann hypothesis, also known as the Riemann conjecture, is a conjecture about the zero distribution of the Riemann zeta function ζ(s).
Based on the fact that the frequency of prime numbers is closely related to the properties of a carefully constructed so-called Riemann zeta function ζ(s), the Riemann hypothesis asserts that all meaningful solutions to the equation ζ(s) = 0 lie on a straight line.
Speaking of which, the birth of the Riemann hypothesis is also something worth pondering.
In 1859, Riemann was elected as a corresponding member of the Berlin Academy of Sciences, for which he submitted a paper.
The title of the paper is "On the Number of Prime Numbers Less Than a Given Value", and the content of the paper is only eight pages long.
One of the major achievements in these eight pages is the discovery that the characteristics of prime number distribution are contained in a special function.
In particular, the series of special points that make this special function take the value of zero have a decisive influence on the detailed laws of prime number distribution.
This special function is now called the Riemann zeta function, and that series of special points are called the non-trivial zeros of the Riemann zeta function.
The interesting thing is that these short eight pages can reflect such significant achievements.
Riemann made the text concise to an excessive degree, and did not express any of the parts that were omitted in the proof.
But the key point is that the parts omitted in these proofs did not allow other mathematicians to make such obvious proofs.
Instead, it took decades of effort by later mathematicians to complete the proof.
And it is not a complete completion yet, some places are still blank to this day.
What is even more interesting is that, in addition to the parts omitted in the proof, Riemann specifically explained a proposition that he explicitly admitted that he himself could not prove.
And this proposition is now the Riemann hypothesis, also known as the Riemann conjecture.
As a result, since its birth, this paper has been like a towering peak in the mathematics world, attracting countless mathematicians to climb to the top.
But after nearly 160 years of research, no one has yet reached the summit.
During this long period of time, although the mathematics community has not solved the Riemann hypothesis, it has produced more than a thousand mathematical propositions.
These mathematical propositions are all based on the premise that the Riemann hypothesis or its generalized form is valid.
If the Riemann hypothesis is proved, then these more than a thousand mathematical propositions will also be promoted to theorems.
On the contrary, if the Riemann hypothesis is falsified, it will cause an earthquake in the mathematical world, and most of the more than one thousand propositions will be buried with the Riemann hypothesis.
But the good news is that the vast majority of mathematicians are optimistic that the Riemann hypothesis will be proved.
Chen Zhou thought the same at this moment.
At least that's what the inspiration he grasped and the collection of wrong answers that recorded his mistakes during the research process told him.
[The Riemann zeta function ζ(s) is the analytical extension of the series expression ζ(s)=∑n=1→∞1/n^s(Re(s)＞1,n∈N＋) on the complex plane]
[Using path integral, the analytically extended Riemann zeta function can be expressed as ζ(s)=Γ(1-s)/2πi∫C(-z)^s/(e ^z-1)dz/z]
Riemann had already completed the analytical extension of this expression, but at that time there was no term "analytical extension" in complex functions.
Chen Zhou looked at the contents written on the draft paper, habitually tapping the draft paper with his pen, and ideas kept flashing in his mind.
He was looking for a breakthrough, relying on the little inspiration he had grasped, to find a breakthrough in the Riemann hypothesis!
The Γ function Γ(s) in the original formula is a generalization of the factorial function on the complex plane. For positive integers s＞1: Γ(s)=(s－1)!
It is obvious that this integral expression is analytic in the entire complex plane except for a simple pole at s = 1, which is also the complete definition of the Riemann zeta function.
Similarly, from this relationship, we can find that the Riemann zeta function satisfies ζ(s)=2^sπ^(s-1)sinπs/2Γ(1-s)ζ(1-s), that is, the Riemann zeta function is zero at s=-2n.
The point on the complex plane where the Riemann zeta function takes the value of zero is called the zero of the Riemann zeta function.
These zero points are distributed in an orderly manner and have simple properties, so they are also called ordinary zero points.
The difficulty lies in that, in addition to these trivial zeros, the Riemann zeta function has many other zeros, whose properties are far more complicated than those of trivial zeros, that is, non-trivial zeros.
A breakthrough idea is needed to prove that all non-trivial zeros of the Riemann zeta function are located on the straight line Re(s)=1/2 in the complex plane, that is, the real part of the solution to the equation zeta(s)=0 is 1/2.
This straight line is also called the critical line by mathematicians!
Suddenly, Chen Zhou put down the pen that was tapping the draft paper, and picked up the draft paper in the BSD hypothesis that made him feel something was wrong.
"The method of half-values and proof by contradiction?"
Chen Zhou muttered to himself, then put the draft paper aside and picked up the pen again.
Time passed by minute by minute...
During the period when Chen Zhou was in seclusion for research, the only thing that aroused discussion in the academic community was probably this year's Nobel Prize ceremony.
However, without Chen Zhou, the discussion became much less heated.
Only Chen Zhou's statement of withdrawing from the selection once again sparked discussion.
Shortly after the Nobel Prize ceremony, Yang Yiyi also went back home for vacation.
However, when she learned that Chen Zhou was in seclusion again, she moved into the hotel reluctantly, and also took on the responsibility of taking care of Chen Zhou's daily catering.
Xiong Hao naturally had no objection to this. It would definitely be much better for Yang Yiyi to take care of Chen Zhou than for him to do so.
By the way, his eating problem can also be solved.
Time quickly came to January 1, 2020, New Year's Day.
In the room, Chen Zhou's face, which was obviously a little tired, had a different kind of glow on it.
"If this is the case, it can also be proved that all non-trivial zeros of the Riemann zeta function are located on the critical line in the complex plane..."
As soon as this thought came to mind, Chen Zhou began to write quickly.
Finally, Chen Zhou solved the Riemann hypothesis that has fascinated countless mathematicians!
Those more than one thousand mathematical propositions will also become true theorems in mathematics as Chen Zhou's proofs are completed!
