0

I would like to know following opinion :

We know that by using Fourier expansion of $f(x)=|x|$ over $0<x<\pi$ one can prove that $\sum_{k=1}^{\infty}\frac{1}{(2k-1)^2}=\frac{\pi^2}{8}$ and thus $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$. Furthermore, by using Fourier expansion of $f(x)=|\cos zx|$ over $-\pi<x<\pi$ , $z\not\in\mathbb{Z}$ and Bernoulli numbers, one can generalize $$\sum_{n=1}^{\infty}\frac{1}{n^{2k}}=\frac{(-1)^{k-1}(2\pi)^{2k}B_{2k}}{2(2k)!}$$ for all positive integers $k$. Here $B_n$ is the $n$-th Bernoulli number.

As you see these are very useful results, show the cool beauty in mathematics. Now I want know that are there any these kind of nice results which can be obtained by Double Fourier Series?? Is there any mathematician who attacked to Riemann hypothesis from this approach??

Jamal Gadirov
  • 111
  • What do you mean by "Double Fourier Series"? Sum of two Fourier series? Product of two Fourier series? Nested sum of an infinite number of Fourier series? There are lots of results that can be obtained from the last one. – Steven Clark May 13 '20 at 18:56
  • @Steven Clark Sir , Fourier Series of $f(x,y)$ on $[-\pi,\pi]\times [-\pi,\pi]$ – Jamal Gadirov May 14 '20 at 09:15

0 Answers0