B Does this help solve the Riemann Hypothesis?

MevsEinstein
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I created an equation with the real part of the input of the zeta function on the RHS and a complex expression on the LHS
Hello PF!

If ##\Re (s)## is the real part of ##s## and ##\Im (s)## is the imaginary part, then t is very easy to prove that $$\zeta (s) = \zeta ( \Re (s) ) \zeta ( \Im (s)i) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}} [\displaystyle\sum_{k \in S, \mathbb{Z} \S = n} \frac{1}{k^{\Im(s)i}}]$$ Since the second term simplifies to ## \zeta ( \Re (s) ) \zeta ( \Im (s)i) - \zeta (s)##. (##\mathbb{Z} \S = n## is actually ##\mathbb{Z}## \##S
## = ##n##). Now, solving for ##\Re (s)## gets us $$\zeta^{-1} ({\frac{\zeta (s) + \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}}[ \displaystyle\sum_{k \in S, \mathbb{Z} \S = n} \frac{1}{k^{\Im (s)i}}]}{\zeta (\Im (s)i)}}) = \Re (s)$$ Setting ##\zeta (s)## to zero gives us the equation we were looking for. Now I am thinking, could this help solve the Riemann hypothesis? Or will this just spit out ##\Re (s)##? I mean, the inverse zeta function is very complicated (page 43 of https://arxiv.org/pdf/2106.06915.pdf ), so it's hard to tell.
 
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The short answer is a simple no.

And there is no such thing as an inverse zeta-function. Let us assume for a second that there is ##\zeta^{-1}.## Then ##\zeta(s)=0## means ##s=\zeta^{-1}(\zeta(s))=\zeta^{-1}(0)##. However, there are infinitely many zeros of the zeta-function. How could that be just a single value ##\zeta^{-1}(0)?## The pre-image ##\zeta^{-1}(\{0\})## is an entire, infinite set. No way to make this a function.

You have to be careful with the zeta-function. E.g., ##\zeta(-1)=-\dfrac{1}{12}## but ##1+2+3+\ldots \neq -\dfrac{1}{12}.## I am not sure what you are doing here so I cannot comment your equations. If you want to even understand the Riemann hypothesis then you should study
https://www.physicsforums.com/insig...thesis-and-ramanujans-sum/#toggle-id-1-closed
 
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To be fair, your first sentence beginning with
I created an equation..
is already alarming and strongly advises to ignore what follows. :frown:
 
MevsEinstein said:
Now, solving for ##\Re (s)## gets us $$\zeta^{-1} ({\frac{\zeta (s) + \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}}[ \displaystyle\sum_{k \in S, \mathbb{Z} \S = n} \frac{1}{k^{\Im (s)i}}]}{\zeta (\Im (s)i)}}) = \Re (s)$$
Ignoring all other possible problems, how is this solving for ##\Re (s)##? It appears on both sides!
 
martinbn said:
how is this solving for ##\Re (s)##? It appears on both sides!
Oops, sorry about that.
 
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