B Does this help solve the Riemann Hypothesis?

AI Thread Summary
The discussion centers on the potential implications of a proposed equation involving the Riemann zeta function for solving the Riemann Hypothesis. It is asserted that the concept of an inverse zeta function is flawed, as the zeros of the zeta function do not correspond to a single value, making it impossible to define a proper inverse. The complexity of the inverse zeta function is emphasized, with caution advised against misinterpretations of its properties. Additionally, there is skepticism regarding the validity of the proposed equations, as they do not clearly solve for the real part of s. Overall, the conclusion is that the approach does not contribute to solving 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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Likes nuuskur, berkeman, malawi_glenn and 1 other person
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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