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:
I am going to speak a little bit about Statistics. The topic today are “averages”. Suppose you have a set of “measurements” where . Then you can define the following quantities:
Remark: In the harmonic mean we need that every measurement is not null, i.e.,
There are some other interesting “averages”/”means”:
Generalized p-th mean.
where are the weight functions and they satisfy
A particularly important case occurs when the weight equals to inverse of the so-called variance of a population with finite size (generally denoted by ), i.e., when , the weighted mean yields:
Finally, a “naive” and usually bad statistical measure for a sample or data set is the midrange. Really, it is a mere measure of central tendency and no much more:
Here, refer to the maximum and minimum value of the sampled variable x in the full data set .
Many of the above “averages” have their own relative importance in the theory of Statistics. But that will be the topic of a future blog post handling statistics and its applications.
What average do you like the most? Are you “on the average”? Are you “normal”? 😉 Of course, you can consult your students, friends or family if they prefer some particular mean/average over any other in their grades/cash sharing, or alike :). See you soon in other blog post!