Would Number Theory Collapse if Riemann's Hypothesis is Proven True?

l-1j-cho
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If Riemann's Hypothesis is proved as true, would number theory collapse?
 
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No. Why would you think that?
 
well, I have absolutely no idea what Riemann hypothesis is, but apparently, if it is proved as true, then we can predict the pattern of prime numbers, of which number theory mostly consists of prime numbers
 
In science, when you get an answer, you get also several new questions.
At the end of 19th century, physicists thought that Physics was nearly terminated.
All the relevant questions had been yet studied and understood.
It only left some "secondary" aspects that would be quickly dispatched...
These secondary points gave birth to both Quantum and Relativistic Physics..

If Riemann's Hypothesis is confirmed, thousands of more interesting questions will arise.
(And if it is refuted, then mathematicians will have a lot of work to do !)

That's why science is so wonderful...
 
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
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