Matematické Fórum

Marian · 06. 04. 2016 08:22

Let s, t be two numbers with s>1 and t>1. Define the functions A(s) and A(s,t) in the following way

$A(s)&:=\sum_{n=1}^{\infty}\frac{1}{n^s},\\ A(s,t)&:=\sum_{n=2}^{\infty}\frac{1}{n^s}\sum_{m=1}^{n-1}\frac{1}{m^t}.$

(a) Prove that for every s, t under consideration the following identity holds true:

$\boldsymbol{A(s,t)+A(t,s)=A(s)\cdot A(t)-A(s+t)}.$

(b) Subsequently, find the closed evaluation of $\boldsymbol{A(2,2)}$ .

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Remark Your solution should be elemental, i.e., only basic facts from the theory of real infinite series are assumed to be known.

vanok · 07. 04. 2016 09:08

Hi ↑ Marian:,
2 lectures
http://eulerarchive.maa.org/hedi/HEDI-2008-01.pdf
https://carma.newcastle.edu.au/MZVs/mzv-week05.pdf

Marian · 07. 04. 2016 10:01

↑ vanok:

Dear vanok,

I'm sufficiently acquainted with the basic theory of multiple zeta values. Please, note that the general reason for my post is not to get materials available on the web, but rather the encouragement of people that are interested in mathematics and, on the other hand, that want to realize your own approachs (succesful or not) or even to construct the whole solution of the problem.

Of course, your URL's can not be treated as a regular solution. Anyway, thank you for your time.

vanok · 07. 04. 2016 19:16

Dear ↑ Marian:,
I put here these URL only to draw attention of the colleagues that it is about a beautiful historic problem dating 1742 (Goldbach, Euler).