The problem:
Is there an ##s\in \mathbb{C}## such that ##\zeta(s)=\zeta(2s)=0##? If so, what is ##\lim_{z\rightarrow s}{\frac{\zeta(s)^{2}}{\zeta(2s)}}##?
Okee dokee, I know this is getting dangerously close to the Riemann Hypothesis, but I swear that wasn't my original intent!
I was playing around with infinite sums and products and I wanted to figure out the sum form of
##\prod_{p}{\frac{1+p^{-s}}{1-p^-s}}##. I started by knowing that
##\prod_{p}{(1-p^{-s})}^{-1} = \zeta(s)##, and from previous work,
##\prod_{p}{\sum_{k=0}^{n-1}{(p^{-s})^{k}}} = \frac{\zeta(s)}{\zeta(ns)}##). From that, I knew
##\prod_{p}{1+p^{-s}} = \frac{\zeta(s)}{\zeta(2s)}##, thus I know the product converges when
##\zeta(2s) \neq 0, \zeta(s) \in \mathbb{C}##. I knew convergence wasn't an issue (for the most part) so I expanded the product some (a reversal of Euler's conversion from
##\zeta(s) = \sum_{k=1}^{\infty}{k^{-s}} = \prod_{p}{(1-p^{-s})}^{-1}##) and obtained:
##\prod_{p}{\frac{1+p^{-s}}{1-p^-s}}= \sum_{k=1}^{\infty}{2^{f(k)}k^{-s}}##,
where
##f(k)## is the number of prime factors of k. It turns out that this has been known since 1979 and after I had access to the internet, I figured out the typical symbol used for my
##f(k)## in this context is
##\omega(k)##. So that was cool, and I could write that sum and product in terms of the zeta function as
##\frac{\zeta(s)}{\zeta(2s)}\zeta(s) = \frac{\zeta(s)^{2}}{\zeta(2s)}##, so do with that what you will (I tried to use it to find the number of prime factors of a number n, but I didn't get anywhere useful). What I decided to pursue was when this function was zero. As long as
##\zeta(s)=0, \zeta(2s)\neq 0##, we know that it is 0, but I haven't yet figured out if there is ever an instance where
##s\in \mathbb{C}, \zeta(s)=\zeta(2s)=0##. If there is such an
s, then the Riemann Hypothesis is false. However, if such an
s does not exist, this says nothing about the Riemann Hypothesis
.
As well, if there is such an
s, I wanted to know if it could be one of those removable discontinuity thingies.
EDIT: fixed a tag and an incorrect restatement of the RIemann Hypothesis
