On upper and lower fast Khintchine spectra of continued fractions

Let psi : N -> R+ be a function satisfying phi(n)/n -> infinity as n -> infinity. We investigate from a multifractal analysis point of view the growth rate of the sums Sigma(n)(k=1) log a(k) (x) relative to psi(n), where [a(1) (x), a(2) (x), ...] denotes the continued fraction expansion of an irrational x is an element of (0, 1). For alpha is an element of [0, infinity], the upper (resp. lower) fast Khintchine spectrum is considered as a function of alpha which is defined by the Hausdorff dimension of the set of all points x such that the upper (resp. lower) limit of 1/psi(n) Sigma(n)(k=1) log a(k) (x) is equal to alpha. These two spectra have been studied by Liao and Rams (2016) under some restrictions on the growth rate of psi. In this paper, we completely determine the precise formulas of these two spectra without any conditions on psi.