Let 1<s<2, and λk>0 with λk→∞ satisfy the Hadamard condition λk+1/λk≥λ>1. For a class of Besicovich functions, B(t)=Σ∞ k=1 λk-2 k sin(λkt), the present paper investigates the intrinsic relationship between box dimension ofgraphs of their vth fractional integrals g(t) and uth fractional derivatives(t) and the asymptotic behavior of{λk}. We show that: if 0<υ<1, s>1+υ, then for sufficiently large λ, dimBГ(g)=dimBГ(g)=s-υ holds if and only if lim n→∞logλkn+1/logλkn=1; if 0<υ<2-s, then for sufficiently large λ, dimBГ(g)=dimBГ(g)=s-υ holds if and only if lim n→∞logλkn+1/logλkn=1.