  The aim of this paper is twofold. Firstly we present our main discovery
arising from experiments which is the tauberian concept of functions of good
variation (FGV). Secondly we propose to use these FGV for proving RH is true
via some conjectures. More precisely we give an implicit definition of FGV and
we provide several smooth and nontrivial exemples from experiments. Then using
a conjectured family of FGV approaching the function $x\mapsto x^{-1}\lfloor
x\rfloor$ we derive RH is true. We make also a tauberian conjecture allowing us
to prove RH is true for infinitely many $L$-functions and we discuss the linear
independance conjecture. The method is inspired by the Ingham summation process
and the experimental support is provided using pari-gp
