I was encouraged to post this here instead of Facebook, which is probably an excellent idea.
Anyways, I was doing some math and I came up with a sum that involved prime factors and such, expressed in terms of the Riemann Zeta function. It got me curious to look at if ζ(s)=ζ(2s)=0 for any s. If yes, then then the Riemann Hypothesis is false (so I am inclined to believe that no such complex s exists). If RH is true, then that would say that there isn't an s, but that might be the more difficult way of proving no such s exists.
So, does anybody have any ideas or approaches to showing an s does or does not exist?
EDIT:
For the interested, I was very originally looking back at a solution to a problem I found in a journal and on my way to the solution, I gave a generalized form for ∏(1+p^-s+p^(-2s)+...+p^(-ks)) as ζ(s)/ζ(ks+s) or something (I don't have the notebook here and I am rushed), and I was looking at ∏(1+p^-s)ζ(s), expanding the zeta function as a sum, which gives a sum similar to the zeta function, but with the n-th coefficient as a function of the number of prime factors of n. From the formula above:
∏(1+p^-s)ζ(s)=ζ(s)^2/ζ(2s)
I just wanted to make sure that whenever the numerator was 0, the denominator wouldn't be (or if it is, hopefully it is something like a simple pole that can be factored out).
Okay, gotta go to work, have fun!
Anyways, I was doing some math and I came up with a sum that involved prime factors and such, expressed in terms of the Riemann Zeta function. It got me curious to look at if ζ(s)=ζ(2s)=0 for any s. If yes, then then the Riemann Hypothesis is false (so I am inclined to believe that no such complex s exists). If RH is true, then that would say that there isn't an s, but that might be the more difficult way of proving no such s exists.
So, does anybody have any ideas or approaches to showing an s does or does not exist?
EDIT:
For the interested, I was very originally looking back at a solution to a problem I found in a journal and on my way to the solution, I gave a generalized form for ∏(1+p^-s+p^(-2s)+...+p^(-ks)) as ζ(s)/ζ(ks+s) or something (I don't have the notebook here and I am rushed), and I was looking at ∏(1+p^-s)ζ(s), expanding the zeta function as a sum, which gives a sum similar to the zeta function, but with the n-th coefficient as a function of the number of prime factors of n. From the formula above:
∏(1+p^-s)ζ(s)=ζ(s)^2/ζ(2s)
I just wanted to make sure that whenever the numerator was 0, the denominator wouldn't be (or if it is, hopefully it is something like a simple pole that can be factored out).
Okay, gotta go to work, have fun!