My second post this day is a beautiful relationship between the Riemann zeta function, the unit hypercube and certain multiple integral involving a “logarithmic and weighted geometric mean”. I discovered it in my rival blog, here:
First of all, we begin with the Riemann zeta function:
Obviously, diverges (it has a pole there), but the zeta value in and can take the following multiple integral “disguise”:
Moreover, we can even check that
In fact, you can generalize the above multiple integral over the unit hypercube
and it reads
I consulted several big books with integrals (specially some russian “Big Book” of integrals, series and products or the CRC handbook) but I could not find this integral in any place. If you are a mathematician reading my blog, it would be nice if you know this result. Of course, there is a classical result that says:
but the last boxed equation was completely unknown for me. I knew the integral represeantations of and but not that general form of zeta in terms of a multidimensional integral. I like it!
In fact, it is interesting (but I don’t know if it is meaningful at all) that the last boxed integral (2) can be rewritten as follows
where I have defined the weight function
and the geometric mean is
and the volume element reads
I love calculus (derivatives and integrals) and I love the Riemann zeta function. Therefore, I love the Zeta Multiple Integrals (1)-(2)-(3)-(4). And you?
PS: Contact the author of the original multidimensional zeta integral ( his blog is linked above) and contact me too if you know some paper or book where those integrals appear explicitly. I believe they can be derived with the use of polylogarithms and multiple zeta values somehow, but I am not an expert (yet) with those functions.
PS(II): In math.stackexchange we found the “proof”:
Just change variables from to and let . For , we have: