Chapter 356 (2/2)
If this is one of the starting points of ∑ - S = (p-1) of the complex number.
This is a rather cold mathematical formula, which is almost difficult to use in the current mathematical academic research.
I didn't expect that President Wei would suddenly think that he would use it as the other entry point to prove Bertrand's hypothesis, which is indeed worthy of being a great bull in the Chinese mathematical circle. It's just that the results don't seem perfect.
It took more than ten minutes for Cheng Nuo to read the whole paper.
Of course, this does not mean that Cheng Nuo has read the complete 34 pages of the document.
As with Cheng Nuo's graduation thesis, it's only the five or six pages.
After reading it, Cheng Nuo understood Wei's idea of proof.
First, he let f (n) be a function satisfying f (N1) f (N2) = f (N1N2), and ∑ n | f (n) | ∞ (N1 and N2 are natural numbers), then we can deduce that ∑ NF (n) = Πp [1 + F (P) + F (P2) + F (P3) +...].
After getting the above series of derivation theorems, it is the first step of proof.Next, because ∑ n | f (n) ∞, 1 + F (P) + F (P2) + F (P3) +... Is absolutely convergent. Consider the part of PN in continuous product (finite product) By using the product property of F (n), it is obtained that Π PN [1 + F (P) + F (P2) + F (P3) +...] = ∑ 'f (n).
In the third step, because 1 + F (P) + F (P2) + F (P3) +... = 1 + F (P) + F (P) 2 + F (P) 3 +... = [1-f (P)] - 1
Step four
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Finally, let (2n)! (n! N!) = Πp ≤ 2n3ps (P). In this paper, the continuous multiplication is divided into two parts: P ≤√ 2n and √ 2np ≤ 2n3 Therefore, Bertrand hypothesis is proved to be true.
Step by step, logical.
The idea is clear, but it seems to be in common sense.
After reading the first time, Cheng Nuo did not find any flaws in the paper.
Cheng Nuo frowned slightly.
Sure enough, it's not that simple.
Cheng Nuo didn't have time to go through it again. He first ruled out the simple part of logical deduction in the paper and ignored it directly.
If that logic error really appeared in that kind of low-level logic deduction step, Wei Yuan Yuan Yuan could not have regarded it as Cheng Nuo's thesis defense topic.
Because, that's embarrassing.
In this paper, there are five places where there are huge amount of calculation and careful derivation steps.
Cheng Nuo investigated one by one.
”The first part is the reasoning of the sum of the right end of the Euler product formula and the ordinary finite product. First, all the f (n) terms containing factor 2 at the right end of the equation are eliminated, and then...”
Second, the distribution of prime numbers and the two-step accuracy
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”Fourth, the substitution of the properties of F (n), f (2) ∑ NF (n) = f (2) + F (4) + F (6) +...
suddenly, Cheng Nuo, who saw this part of the content, suddenly had a glance.
He looked at the formula and looked at the corner of his mouth.
I found you!
Cheng Nuo picked up a carbon pen and wrote on the sketch paper for a while. Then he drew a horizontal line under the formula of the paper.
The formula on the horizontal line: Πp [1-f (P)] ∑ NF (n) = f (1) = 1, (2n)! (n! N!) = Πp ≤√ 2nps (P), ∑ NF (n) = Πp [1-f (P)] - 1
that's where it is, right.
There is a habitual error in the logical relationship between the third formula and the first two formulas.
These three formulas are one of the core formulas in the proof process of the whole paper. Therefore, the error of the formula leads to the whole paper becoming a waste of paper.
Cheng Nuo is in a very good mood at this time.
Because he not only found the logical error required by President Wei, but also worked out a reasonable correction plan in his mind!
Looking up, there was no one on the reply table in front of the four teachers.
Cheng Nuo picked up the paper and swaggered onto the platform.
A teacher's eyes, then a faint smile