The most beautiful chain of equalities I have ever seen

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In summary: That will get you started.In summary, a surprising equality between a sum and a product was discovered while analyzing the Dedekind eta function and Dirichlet series. This suggests a deep connection between prime numbers and chaos, which has been further explored by many mathematicians. Further results have shown that this relationship also extends to the golden ratio and other functions.
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epr1990
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I was doing some basic analysis of the Dedekind eta function and some Dirichlet series and the following equality just fell out:

[tex]\sum_{k=1}^\infty\frac{\mu (k)-\varphi (k)}{k}\log \left( 1-\frac{1}{\phi^k} \right) = \prod_{k=1}^\infty \left( 1-\frac{1}{\phi^k} \right)^{2\pi i\frac{\mu (k)-\varphi (k)}{k}}[/tex]

where [itex]\mu , \varphi , \phi [/itex] are the Möbius function, Euler totient function, and golden ratio respectively.

Now, at first, this looks almost nonsensical because it demonstrates equality between a product and its logarithm. I.e. exponentiating gives product=e^product^2*pi*i (or you can take the logarithm if you can see it better this way, personally, I'm better with products than sums, and definitely the logarithm of an infinite sum which you would most likely need to refactor and deal with possible rearrangement issues... yikes). It follows that, in a sense, this form is redundant, for we must have

[tex]\left(\prod_{k=1}^\infty \left( 1-\frac{1}{\phi^k} \right)^{\frac{\mu (k)-\varphi (k)}{k}}\right)^{2\pi i} = 1 [/tex]

Furthermore, this implies

[tex]\sum_{k=1}^\infty\frac{\mu (k)-\varphi (k)}{k}\log \left( 1-\frac{1}{\phi^k} \right) = 1[/tex]

and

[tex]\prod_{k=1}^\infty \left( 1-\frac{1}{\phi^k} \right)^{\frac{\mu (k)-\varphi (k)}{k}} = e[/tex]

Finally, using Dirichlet convolution and inversion and the basic properties of the logarithm and geometric series, we can show that
for 0 < x < 1

[tex]\sum_{k=1}^\infty\frac{\mu (k)}{k}\log \left( \frac{1}{1-\frac{1}{x^k}} \right) = x[/tex]

and

[tex]\sum_{k=1}^\infty\frac{\varphi (k)}{k}\log \left( \frac{1}{1-\frac{1}{x^k}} \right) = \frac{x}{1-x}[/tex]

Thus,

[tex]\sum_{k=1}^\infty\frac{\mu (k)}{k}\log \left( \frac{1}{1-\frac{1}{\phi^k}} \right) = \frac{1}{\phi}[/tex]

and

[tex]\sum_{k=1}^\infty\frac{\varphi (k)}{k}\log \left( \frac{1}{1-\frac{1}{\phi^k}} \right) = \frac{\frac{1}{\phi}}{1-\frac{1}{\phi}} = \phi[/tex]

The point of this result is that it suggests a deep relation between the prime numbers and chaos,


I was hoping that anyone who reads this would share their thoughts and insights on this relationship, or possibly similar results.
 
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Sure - google for "prime numbers and chaos math".
 

Related to The most beautiful chain of equalities I have ever seen

1. What is "The most beautiful chain of equalities I have ever seen"?

"The most beautiful chain of equalities I have ever seen" is a phrase coined by physicist Richard Feynman to describe a mathematical proof in quantum electrodynamics.

2. What makes this chain of equalities beautiful?

The beauty of this chain of equalities lies in its simplicity and elegance. It encompasses fundamental principles of physics and explains complex phenomena in a concise and elegant manner.

3. What is quantum electrodynamics?

Quantum electrodynamics is a theory that describes the interactions between charged particles and electromagnetic fields. It is a cornerstone of modern physics and has been extensively tested and verified through experiments.

4. Who discovered this chain of equalities?

The chain of equalities was discovered by Richard Feynman, a renowned theoretical physicist and Nobel laureate. He developed this proof as part of his work in quantum electrodynamics in the 1940s.

5. How has this chain of equalities influenced modern physics?

This chain of equalities has had a significant impact on modern physics, particularly in the field of quantum field theory. It has provided a framework for understanding and predicting the behavior of particles and fields at the quantum level, and has been essential in the development of other theories like the Standard Model of particle physics.

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