Author's: ALI H. HAKAMI
Pages: [39] - [52]
Received Date: September 12, 2012
Submitted by:
Let m be a positive integer. Denote to the ring by
and a Cartesian product of n copies
of
by
Let
be a quadratic polynomial in
Write
where
and
is a quadratic form given by
where A is a symmetric
matrix with integer entries. Assume
unless we mention else. Let V be
the set of points in
satisfying the congruence
If
and
we shall say
if x is congruent to y
modulo the ideal
For any subset S of
and divisor d of m, let
where
denote to the cardinality. Let
denote the Euler phi-function,
denote the number of distinct positive
divisors of m, and for positive integers
set
In this paper, we shall prove that for any
subsets S and T of
with
we have
We also show that the above result can be made more precise when
is a box of points in