  The previous attempts of reconstructing the Gell-Mann-Low function \beta(g)
of the \phi^4 theory by summing perturbation series give the asymptotic
behavior \beta(g) = \beta_\infty g^\alpha in the limit g\to \infty, where
\alpha \approx 1 for the space dimensions d = 2,3,4. It can be hypothesized
that the asymptotic behavior is \beta(g) ~ g for all values of d. The
consideration of the zero-dimensional case supports this hypothesis and reveals
the mechanism of its appearance: it is associated with a zero of one of the
functional integrals. The generalization of the analysis confirms the
asymptotic behavior \beta(g)=\beta_\infty g in the general d-dimensional case.
The asymptotic behavior of other renormalization group functions is constant.
The connection with the zero-charge problem and triviality of the \phi^4 theory
is discussed.
