Zeta 3

Note: I provide this tutorial with no guarantee of its accuracy. It is here for your own enjoyment. If you would like to use it in your own papers or theories, the work should be checked and a refrence to the author would be nice.

The Zeta[3] is a transcendental number that no one knows what its value is exactly. It is equal to the infinite sum It may be that we have no other way of representing it than we do as an infinite series. However, many do not want to except it (even if it is inevitable).

Although I doubt it is new, it is my hope that this will aid someone in to finding any exact answer.

I started off using an integral that I knew would yield Zeta[3].

Analyzing the graph, I could see that a shift in the graph would yield the same result if I doubled it. Also, this would make the lower boundry zero so we would only have to evaluate the integral at one point.

Therefore, by shifting the graph to the left 1/2 we would get exactly 1/2 the value of the Zeta[3]. So, by replacing the the x with x-1/2 and simplifying, we get a slightly more simple integral.

Here, I proceded to convert the integrand into a series in hopes that I could decompose the series into seperate series and find the value that way.

With a bit of work, I was able to determine that the general term to this.Note that the B is the BernoulliB number.

Now, we integrate this general term to get this:

So now we plug in x=1/2 and take the sum from 1 to infinity. To do this, we need to know what this term is at 1. To do this, we take the limit of the general term.

Alright, now we the term at 1. Now, I split up the general term and took the series from 2 to Infinity. The split up terms with x->1/2 is below.

Now we take the sums.

Well, as promising as this seems, Mathematica takes the third and forth sums (which I've combined below) in terms of Zeta[3].

However, perhaps someone can take those 2 sums (the first two terms above) into something else. Anyway, combining everything, the final equation is below.

Well, there you have it! I certainly hope this has provided some insight and that I have not skipped over anything too vital. Let me know if this helps you or if you have questions!