Does Euler's Product Over Primes Converge for s > 1?

tpm
Messages
67
Reaction score
0
Following Euler if we define the product:

(x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s})...=f(x)

taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that f(x,s)=1/Li_{s} (x) (inverse of Polylogarithm) however I'm not 100 % sure, although for x=1 you get the inverse of Riemann Zeta
 
Last edited:
Physics news on Phys.org
1. Inverse and Reciprocals aren't the same thing.

2. For any value of s, and x > 1, the terms do not approach 1, but x. x being more than 1, the terms are not approaching 1, so the product does not converge.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\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...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

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...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

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...
View full post »

Similar threads

I Extend Euler Product Convergence Over Primes: Basic Qs
Replies
0
Views
1K
I Meaning of "identify" & variations in different math context
Replies
5
Views
510
I Meaning of "passage to the quotient"?
Replies
3
Views
448
I Passive Transformation and Rotation Matrix
Replies
1
Views
3K
I Connection of zeta to primes: less Euler than error or L-functions?
Replies
2
Views
2K
I No structure in ##(d,x)## in ##\textbf{Met*}(X)## is admissible
Replies
0
Views
1K
Is this a route to the prime number theorem?
Replies
1
Views
2K
Is There a Way to Regularize Euler Products on Primes?
Replies
1
Views
3K
MHB Tensor Products - D&F - Extension of the scalars
Replies
2
Views
1K
Primes as roots of same function
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top