  In this article, we obtain a class of tame maximal weights (Zhou weights).
Using Tian functions (the function of jumping numbers with respect to the
exponents of a holomorphic function or the multiples of a plurisubharmonic
function) as a main tool, we establish an expression of relative types (Zhou
numbers) to these tame maximal weights in integral form, which shows that the
relative types satisfy tropical multiplicativity and tropical additivity. Thus,
the relative types to Zhou weights are valuations (Zhou valuations) on the ring
of germs of holomorphic functions. We use Tian functions and Zhou numbers to
measure the singularities of plurisubharmonic functions, involving jumping
numbers and multiplier ideal sheaves. Especially, the relative types to Zhou
weights characterize the division relations of the ring of germs of holomorphic
functions. Finally, we consider a global version of Zhou weights on domains in
$\mathbb{C}^n$, which is a generalization of the pluricomplex Green functions,
and we obtain some properties of them, including continuity and some
approximation results.
