In [BI3], we introduced the notion of ratio Mercerian theorem, to improve the Mercerian theorem for Fourier and Hankel transforms first proved in [BI1]. In the first half of this paper, we extend (and correct) the ratio Mercerian theorem, and apply it to the proofs of Tauberian theorems for kernels of Korenblum type. The latter can be applied to analytic number theory; this was in fact the moti...