Distribuion of Leading Digits of Numbers II

In this paper, we study the sequence (f (p_n))_n≥1,where p_n is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=log_b x^r and f (x)=log_b(x log x)^r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy DNi*(f(pn),g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (N_i)_i≥1 of the natural numbers. Especially for f (x)= log_b x^r, we obtain the effective results for an upper bound of DNi*(f(pn)g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).