For a finite system
$
A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}}
$
A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}}
of arithmetic sequences
the covering function is
w(
x)
= |{1
$
{\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } }
$
{\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } }
. (b) If
n1
$
{\left( {{*{20}c}
{l} \\
{r} \\
} \right)}
$
{\left( {\begin{array}{*{20}c}
{l} \\
{r} \\
\end{array} } \right)}
can be written as the
sum of some (not necessarily distinct) prime divisors of
nk. (c)
max
(x$
{\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } }
$
{\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } }
where
m1, .
. .,mk are positive
integers.Mathematics Subject
Classification (2000): 11B25 - 05A15 - 11A07 - 11A25 - 11B75 - 11D68
The research is supported by the Teaching and
Research Award Fund for Outstanding Young Teachers in Higher
Education Institutions of MOE, and the National Natural Science
Foundation of P. R. China.