Hi everyone, I am an undergraduate engineering student really interested in mathematics and lately I have just been playing around with a few ideas that I would like a second opinion (possible fact check) on.
After reproving (with some help) the Basel problem using Euler’s method of expanding sinx/x into an infinite product and comparing the terms in the Taylor series expansion I figured that we could do a similar method to find zeta(3). It seems as if I am able to just create a function that is in the same form of an infinite product only there is an n3 in the denominator of the one term instead of n2. It does not have to be exactly similar to (1-x2 /(npi)2 )but just be of the form (1 + xa /(n3 )) and then depending on the integer a used I can just find zeta(3) in terms of the a’th derivative of this function. The only problem is that I do not have an actual function to calculate these derivatives to then expand into Taylor series. Why is it that there seems to be no way to find such a function (as I have obviously been unsuccessful)?
My next question deals with a paper I read that detailed very accurate approximations of zeta(3) assuming it was of the form a*pi3 where a is some rational number. They were able to get very close but deduced that it is most likely not of this form considering they got up to a rational number with hundreds of digits in the denominator and numerator and still had some error. They also tested other forms involving radicals and ln(2) terms as well all having the same problem lending to the possibility that zeta(3) having a VERY complicated closed form, or possibly none at all.
This brings me to my last question. Is it really possible that zeta(3) is a standalone constant we should just assign another symbol/greek letter to (or just leave as zeta(3))? Or can it in fact be expressed in terms of other known constants in some coveted “closed form” expression. But what really is a closed form expression then? This got me thinking of how the set of natural numbers and integers have this property that any and all members of these sets can be expressed in terms of a finite amount of algebraic operations of other members of the set. I am not sure what this property is called (why does this feel a lot like linear independence type stuff in linear algebra?). I know and understand that real numbers and therefore by extension transcendental numbers are uncountable (I have seen the proofs) so does this mean that these sets do not have this property? I guess it would make sense considering this “ability to express others in a finite amount of ALGEBRAIC operations” wouldn’t apply to transcendental numbers since they are literally defined to be NOT algebraic. We also don’t even know if zeta(3) is transcendental...
But let’s say that the transcendental numbers did have this property and we are able to prove that zeta(3) is transcendental. If we were able to prove that the transcendental numbers have this property than that means a “closed form” zeta(3) could be stumbled upon in the same manner the previous paper used to find approximations to it. Wouldn’t proving this property also help us prove the transcendence of numbers like pi+e and pi*e?
I lied I have one last question. It says on the wikipedia page that the zeta function at odd values isn’t even known to be irrational (zeta(3) is though), yet I could have swore I saw a paper claiming to prove zeta(2n+1)’s irrationality. Is this maybe very recent, or just incorrect? Or maybe the wikipedia page isn’t checked as this was deep within an article not really pertaining to the zeta function.
Anyways, thank you!
tl;dr Why can’t we use an Eulerian method for finding zeta(3)?
What are the implications of proving zeta(3)’s likely transcendence and is the property discussed above only true for countable sets (naturals, rationals)? And if not, would proving this property for transcendentals possibly allow us to find zeta(3) or prove pi+e as transcendental?
Is it possible that zeta(3) has no “closed form,” or in other words cannot be expressed in terms of a finite amount of algebraic operations of other known constants like pi, e, ln(2), etc.?
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from math https://www.reddit.com/r/math/comments/gsnmpt/transcendental_numbers_apérys_constant_and_an/
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