Affiliation:
1. Dept. of Businessv Administration , The International University of Kagoshima , Kagoshima , Japan
2. Institue of Mathematics, Slovak Academy of Sciences , Bratislava , Slovakia
Abstract
Abstract
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
D
N
i
*
(
f
(
p
n
)
,
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
D
N
i
*
(
f
(
p
n
)
g
)
D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right)
.
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