Unsolved problems in Fourier analysis

Question

I am interested in the classical theory of Fourier analysis and would like to ask if anyone could name a few unsolved problems in this field, and especially, in the classical study of Fourier series?

Answer 1

Some well-known mathematical problems can be stated in term of Fourier analysis :

The coefficients of the sine wave in the sawtooth basis encodes everything about the Riemann hypothesis. More precisely, let $s(x) = \lfloor x \rfloor+\frac12-x = \sum_{n=1}^\infty \frac{\sin(2 \pi n x)}{\pi n}$ the saw-tooth.

Then $$\frac{\sin(2 \pi x)}{\pi }=\sum_{n=1}^\infty \frac{\mu(n)}{n} s(nx)$$ where $\mu(n)$ is the Möbius function, and the rate of convergence of this series determinate the truth of the Riemann hypothesis.