Karlheinz Gröchenig
We review Schoenberg's characterization of totally positive functions and its connection to the Laguerre-Polya class. This characterization yields a new condition that is equivalent to the truth of the Riemann hypothesis.
Karlheinz Gröchenig
This paper contains 10 theorems, 34 equations.
Theorem 1
(i) If $\Lambda$ is a Polya frequency function, then its (two-sided) Laplace transform converges in a vertical strip $\{z\in \mathbb{C} : \alpha < \mathrm{Re}\, z < \beta \}, \alpha <0< \beta$, and is the reciprocal of a function $\Psi$ in the Laguerre-Polya class with $\Psi (0)>0$. (ii) Conversely, if $\Psi$ is in the Laguerre-Polya class with $\Psi (0) >0$, then its reciprocal $1/\Psi$ is the L
