Differential equations properties and $\psi$ function
Résumé
Using the differential equations properties, we study the function $\psi$ which is a function defined by an integral. The particular case of this function coincide with the function $s\mapsto \zeta(s)/s$, where $\zeta$ is the Riemann Zeta function. In the end of the paper, we find that $\frac{\zeta(s)}{s}\neq -\frac{\zeta(1-s)}{1-s}$ for every $\Re(s)\in(0,\frac{1}{2})$ and $\Im(s)\in \mathbb{R}^*$. Thanks to the Riemann functional equation, we deduce that any "non-trivial zero" of $\zeta$ is of real part equal to $\frac{1}{2}$.
Domaines
Mathématiques générales [math.GM]
Origine : Fichiers produits par l'(les) auteur(s)