Let
N
0
be a integral ideal divisible by 4, of a totally real field
K
. We show that there is the Shimura lifting map of a space of Hilbert modular forms with character modulo
N
0
of half-integral ...weight, to the space of Hilbert modular forms of integral weight under some condition. In particular it is shown that if
16
|
N
0
, then any Hilbert modular forms of weight at least 5 / 2 has the Shimura lift. As an application, we compute the Shimura lifts of the third powers of theta series for
K
=
Q
(
2
)
and
K
=
Q
(
5
)
, and obtain the formulas for the numbers of representations of totally positive integers in
K
as sums of three integral squares.
Let
denote the set consisting of those integers which can be written as sums of three squares. We prove that if 0 ≤
k
≤
n
and
, then
k
≤ 73. We then study how many consecutive binomial coefficients ...may belong to
, and prove that for any given
k
we can find infinitely many values of
n
such that at least
k
consecutive coefficients
belong to
. We also prove the existence of infinitely many quadruples of consecutive binomial coefficients that cannot be written as sums of three squares and that from five consecutive binomial coefficients at least one is a sum of three squares.
For each nonnegative integer n, r_3(n) denotes the number of representations of n by sums of three squares. Here presented is a two-step recursive scheme for computing r_3(n), n ≥ 0.
Let
s
(
n
)
be the number of representations of
n as the sum of three squares. We prove a remarkable new identity for
s
(
p
2
n
)
−
p
s
(
n
)
with
p being an odd prime. This identity makes nontrivial ...use of ternary quadratic forms with discriminants
p
2
,
16
p
2
. These forms are related by Watsonʼs transformations. To prove this identity we employ the Siegel–Weil and the Smith–Minkowski product formulas.
For a positive integer k and a certain arithmetic progression A, there exist infinitely many quadratic fields \mathbf{Q}(\sqrt{-d}) whose class numbers are divisible by k and d\in A. From this, we ...have a linear congruence of the representation numbers of integers as sums of three squares.
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