Let Ξ³nβ (nβZβ₯0β) be a sequence of complex numbers,
which is tame: 0<βuβ€Ξ³nβ1β/Ξ³nββ€βv<β for
all n>0. We show a resonance between the singularities of the function of the
power series P(t):=βn=0ββΞ³nβtn on its boundary of the disc
of convergence and the oscillation behavior of the sequences
Ξ³nβkβ/Ξ³nβ (nβZ>>0β) for k>0. The resonance is
proven by introducing the space of opposite power series, which is the compact
subspace of the space of all formal power series in the opposite variable
s=1/t and is defined as the accumulating set of the sequence
Xnβ(s):=βk=0nβΞ³nβΞ³nβkββtk (nβZβ₯0β). We analyze in details an example of the growth series
P(t) for the modular group PSL(2,Z) due to Machi.Comment: 25 page