Consider an end Ω in the sense of Heins [7]: Ω is a relatively noncompact subregion of an open Riemann surface R, which is of null boundary and has a single ideal boundary component, and the relative boundary dΩ(^ 0) of Ω consists of finitely many analytic closed Jordan curves. We denote by P(Ω) the class of nonnegative harmonic functions on Ω with vanishing boundary values on dΩ. The minimum number of elements o f P ( Ω ) generating P(Ω) provided that such a finite set exists, otherwise oo, is referred to as the harmonic dimension of β, dimP(Ω) in notation. In terms of Martin compactification, it is known that dίmP(Ω) coincides with the number of minimal boundary points in the Martin compactification of R (cf. e.g. [4]), and hence dimP(Ω) = dimP(Ω') for any pair (β, Ω') of ends of R. Denote by D the punctured unit disc {0 < \z < 1} and let be a p-sheeted (1 < p < oo) unlimited covering surface of D such that the projection of branch points of accumulates only at z = 0. Then is naturally considered as a subregion of an open Riemann surface R which is a p-sheeted unlimited covering surface of {0 < \z < oo}. I f R has a single ideal boundary component, it is seen that is an end. We denote by ep the class of ends of this kind. In this paper we are especially concerned with ends belonging to £p. First of all, it is noted that 1 < dimP(W) < p for each € £p (cf. Heins [7]). Roughly speaking, if each sheet of is closely connected with any of the other sheets, then dίmP(W) = I and if each sheet of is faintly connected with the other sheets, then dimP(W) — p. This intuition is realized as follows. Consider two positive decreasing sequences {αn} and {bn} satisfying 6n+ι < an < bn < I and lim^^ an = 0. Set G = {0 < \z < 1} /, where / = U^=lln and In = [αn, bn]. We take p copies G\, , Gp of G and join the upper edge of In on Gj with the lower edge of In on Gj+i (j mod p) for every n. Then we obtain a p-sheeted covering surface W of D which belongs to 8P. For this end W we have showed the following (cf. [9], [11] and [14]).