🌟 Exciting News! 🌟
I'm thrilled to share some amazing updates featuring the incredible #AshleyJudd! Whether you're a fan of #Allegiant or #Insurgent, you won't want to miss this! Stay tuned for more and let's celebrate the talent that brings our favorite characters to life!
🌟 Dive into the world of Divergent with two amazing stars! 🌟
#ZoeKravitz and #Christina bring the captivating character of Christina to life in #TheDivergent series! Both their performances are absolutely unforgettable. Have you watched it yet? Let us know your favorite scene! 🎬✨
Riemann zeta function \(\zeta(s)\) and \(\displaystyle\sum_{n=1}^\infty n=1+2+3+\cdots=-\dfrac{1}{12}\)
Have you ever heard that the sum of all natural numbers is \(-1/12\)?🤔 Of course not; this doesn't make sense in the usual sum, but using a summation method based on analytic continuation of the Riemann zeta function leads to the following result.
The Riemann zeta function is defined as:
\[\zeta(s)=\displaystyle\sum_{n=1}^\infty\dfrac{1}{n^s}=\dfrac{1}{1^s}+\dfrac{1}{2^s}+\dfrac{1}{3^s}+\cdots\]
for \(s\in\mathbb{C}\) such that \(\Re(s)>1\).
It can be extended to a meromorphic function with only a simple pole at \(s=1\), using analytic continuation and the following functional equation:
\[\zeta(1-s)=2^{1-s}\pi^{-s}\cos\left(\dfrac{\pi s}{2}\right)\Gamma(s)\zeta(s)\]
For \(s=2\), this gives \(\zeta(-1)=\displaystyle\sum_{n=1}^\infty n=-\dfrac{1}{2\pi^2}\zeta(2)=-\dfrac{1}{2\pi^2}\cdot\dfrac{\pi^2}{6}=-\dfrac{1}{12}\), which is a reason for assigning a finite value to the divergent sum/series (zeta function regularization). That is, \(\displaystyle\sum_{n=1}^\infty n=1+2+3+\cdots=-\dfrac{1}{12}\).
#RiemannZetaFunction #ZetaFunction #Riemann #DivergentSum #DivergentSeries #FiniteValue #ZetaRegularization #ZetaFunctionRegularization #NegativeFraction #MeromorphicFunction #AnalyticContinuation
@charliewood on Jean Écalle’s resurgence theory—how to go beyond the pertubation theory of divergent series in quantum field theory, to the underlying exact result—
https://www.quantamagazine.org/alien-calculus-could-save-particle-physics-from-infinities-20230406/
I can’t remotely keep up with this, but it strikes me as a very big deal.
In the math of particle physics, every calculation should result in infinity. Physicists get around this by just ignoring certain parts of the equations — an approach that provides approximate answers. But by using the techniques known as “resurgence,” researchers hope to end the infinities and end up with perfectly precise predictions.
I saw something interesting recently linking a bit of #maths with #computing.
The #DivergentSeries 1+2+4+8+᠁ = -1. To the uninitiated, this may seem crazy and completely devoid of meaning. However, look at the two’s complement representation of negative numbers. In particular, if you had a computer with arbitrary precision, you would get the binary representation -1 = 11111᠁ which is also the sum above.
https://en.wikipedia.org/wiki/Two's_complement
https://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_⋯
Rusty Corgi
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