Erdos' Series & Prime Number Theorem Implications

  • Thread starter Thread starter Dragonfall
  • Start date Start date
  • Tags Tags
    Primes Series
Dragonfall
Messages
1,023
Reaction score
5
Erdos noticed that \sum(-1)^n\frac{n\log n}{p_n} diverges, where pn is the nth prime. I can't prove this conclusively. All I can say is that PNT implies that p_n~nlogn and thus the series "resembles" \sum(-1)^n.
 
Physics news on Phys.org
If the terms don't go to zero, then the sum doesn't converge, right?
 
Oh ya, how the hell did I miss that?
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

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 »

Similar threads

I Fundamental theorem of arithmetic from Jordan-Hölder theorem
Replies
5
Views
2K
Proof about a prime between k and 2k.
Replies
6
Views
5K
I Question on Primitive Roots and GCDs
Replies
2
Views
2K
Possible convergence of prime series
Replies
8
Views
4K
A Proving Goldbach's conjecture by induction (or why it doesn't seem to work)
Replies
10
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
Replies
3
Views
2K
Asymptotic formula for the sum of log(p)/p
Replies
2
Views
6K
I Understanding the nth-term test for Divergence
Replies
17
Views
5K
How Is the Square of the Zeta Function Related to the Divisor Function?
Replies
2
Views
4K
Infinite Semi-Pronic Prime Series Appears to Converge
Replies
3
Views
4K

Hot Threads

Recent Insights

Back
Top