Crystalline measures and zeta functions
De Yves Meyer
Apparaît dans la collection : Abel in Paris 2024
A zeta function is associated with any even crystalline measure $\mu$ on $\mathbb{R}$. If $\mathcal{F}(\mu)=\mu$ where $\mathcal{F}$ denotes the Fourier transform this zeta function satisfies the same functional equation as the Riemann zeta function. Does the converse implication hold? Let $\phi(s)=\Sigma^\infty_1 a_k \lambda^{-s}_k$ be a Dirichlet series. One assumes that $\phi(s)$ satisfies the same functional equation as the Riemann zeta function. Does there exist an even crystalline measure $\mu$ which generates $\phi$ and such that $\mathcal{F}(\mu)=\mu$? Kahane and Mandelbrojt addressed this issue. A solution is given in this talk.
