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by **Yablon** » Fri Feb 27, 2015 4:51 pm

Hello to all:

I was always under the impression that nobody has ever found $\pi$ through any exact series. But I was just studying the relationship between Planck's blackbody law and its integration into the Stefan-Boltzman law for totally different reasons (having to do with my present work on thermodynamics) and I discovered that when we derive the Stefan-Boltzmann constant:

$\sigma =\frac{2k_{{\rm B}}^{4} \pi ^{5} }{15c^{2} h^{3} } \approx 5.670400\times 10^{-8} {\rm J}{\rm s}^{{\rm -1}} {\rm m}^{{\rm -2}} {\rm K}^{{\rm -4}}$

by integrating Planck's law (see http://en.wikipedia.org/wiki/Planck%27s_law and http://en.wikipedia.org/wiki/Planck%27s ... tzmann_law), there is built into this derivation the apparent fact that the 4th power polylogarithm of 1 is precisely equal to $\pi^4/90$ , that is:

${\rm Li}_{4} \left(1\right)=\sum _{k=1}^{\infty }\frac{1}{k^{4} } =1+\frac{1}{2^{4} } +\frac{1}{3^{4} } +\frac{1}{3^{4} } +\frac{1}{4^{4} } +\frac{1}{5^{4} } +\cdots =\frac{\pi ^{4} }{90}$ .

Is anyone else aware of this? I am surprised that I was never aware of this.

Jay

by **Yablon** » Fri Feb 27, 2015 9:23 pm

I spent a bit of time figuring out where the 90 comes from, in terms the 4 in ${\rm Li}_{4} \left(1\right)$ . I believe this is the formal characterization of $\pi$ as a series:

$\left(4-1\right)!\cdot {\rm Li}_{4} \left(1\right)=\left(4-1\right)!\sum _{k=1}^{\infty }\frac{1}{k^{4} } =\frac{\pi ^{4} }{2^{4} -1}$

I am curious what numerical numbers X generally result from:

$\left(n-1\right)!\cdot {\rm Li}_{n} \left(1\right)=\left(n-1\right)!\sum _{k=1}^{\infty }\frac{1}{k^{n} } =\frac{X^{n} }{2^{n} -1}$ ,

given that $\pi$ emerges from $n=4$ . I wonder if $e=2.7182818284...$ emerges from some variation of this. Our buddy Ben may have something to say about this.

Jay

by **Yablon** » Fri Feb 27, 2015 9:41 pm

Found it:

http://en.wikipedia.org/wiki/Riemann_ze ... fic_values

So that is where the 15 comes from in the Stefan-Boltzmann constant.

Jay

by **Q-reeus** » Sat Feb 28, 2015 4:04 am

Evidently there has been a lot of work in expressing pi as various power series...http://functions.wolfram.com/Constants/ ... owAll.html
Convergence of such series is not mentioned but a simple one shown for pi^6 will clearly rapidly converge. Then just take the sixth root - 'easy'.

by **Ben6993** » Sat Feb 28, 2015 4:29 am

Jay wrote: .... the 4th power polylogarithm of 1 is precisely equal to : ..... Li4(1) = pi^4/90 ......
Is anyone else aware of this? I am surprised that I was never aware of this.
{not an exact quote}

Hi Jay,
I did not have the coefficient '90' in my memory but it is impossible for me to forget the formula for Li2(1) = pi ^2 / 6 as it is so ingrained by now. As a student I still remember the homework where we had to show that Li2(1) converged. We were probably given some hint to go on to find the exact formula, I can't remember that much as it was nearly fifty years ago. I should imagine that almost every maths graduate knows that formula. You knew about tossing blocks of wood. I knew about pi^2/6 in a similar fashion.

... given that pi emerges from n=4. I wonder if e=2.7182818284 emerges from some variation of this. Our buddy Ben may have something to say about this.

Well, pi emerges from Li2(1) also, and presumably from many others. I do not know the most efficient way of calculating pi [to a given degree of accuracy] from fewest terms in a series. Presumably Li4(1) is more efficient than Li2(1) but I doubt it is the most efficient series.

In terms of being 'exact', if you have a known value of pi at the outset then you shortcut straight to the answer pi^2 / 4 or pi^4 / 90 to solve your infinite series. But, reversing that process, and using the infinite series as a way of finding an exact value of pi ... then you still have to sum more and more and more terms of the infinite series to gradually home in on a more and more and more exact value for pi.

There are many series for e here:
http://en.wikipedia.org/wiki/List_of_representations_of_e

These series seemingly all involve, for e, terms of the type 1/n! rather than the terms of the type 1/n^m which lead to pi.
(Ignoring the well-known formula connecting e, i, pi and -1.)

Found it:
http://en.wikipedia.org/wiki/Riemann_ze%20...%20fic_values
So that is where the 15 comes from in the Stefan-Boltzmann constant.

Some time long ago I did some reading on number theory from A Source Book in Mathematics by D.E.Smith Volumes 1 and 2, Dover, 1959. But it was difficult to find the time to continue with it while employed working in a different area. I also, in the later years, when I first had a home computer, played at reading and using some papers by Euler trying to find a formula to predict the primes using the zeta function. But it is headache territory. If 'he who cannot be named' is a third rate mathematical statistician then I am nth rate one with n>>3.

by **Yablon** » Sat Feb 28, 2015 3:32 pm

Q-reeus wrote:Evidently there has been a lot of work in expressing pi as various power series...http://functions.wolfram.com/Constants/ ... owAll.html
Convergence of such series is not mentioned but a simple one shown for pi^6 will clearly rapidly converge. Then just take the sixth root - 'easy'.

Hi Q-reeus,

Thanks, that is a great link!

Jay

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