Cryptography Reference
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nonpositive integers. It also yields the relation Γ(
n
)=(
n
−
1)! for positive
integers
n
.
COROLLARY 14.5
Let
E
and
N
be as in T heorem 14.4. T hen
(
√
N/
2
π
)
s
Γ(
s
)
L
E
(
s
)=
∓
(
√
N/
2
π
)
2
−s
Γ(2
− s
)
L
E
(2
− s
)
for all
s ∈
C
(an d bothsides continue analyticallytoallof
C
). T he sign here
isthe oppositeofthe sign in (2) of T heorem 14.4.
PROOF
Using the definition of the gamma function, we have
a
n
(
√
N/
2
πn
)
s
∞
0
(
√
N/
2
π
)
s
Γ(
s
)
L
E
(
s
)=
∞
t
s−
1
e
−t
dt
n
=1
a
n
∞
0
=
∞
(
u
√
N
)
s
e
−
2
πnu
du
u
(let
t
=2
πnu
)
n
=1
=
∞
0
(
u
√
N
)
s
f
E
(
iu
)
du
u
=
1
/
√
N
0
+
∞
1
/
√
N
(
u
√
N
)
s
f
E
(
iu
)
du
u
(
u
√
N
)
s
f
E
(
iu
)
du
u
.
(The interchange of summation and integration to obtain the third equality
is justified since the sum for
f
(
iu
) converges very quickly near
∞
.) Let
be
the sign in part (2) of Theorem 14.4. Then
f
E
(
i/
(
Nu
)) =
(
iu
)
2
f
E
(
iu
)=
−u
2
f
E
(
iu
)
.
Therefore, let
u
=1
/N v
to obtain
1
/
√
N
=
−
∞
(
u
√
N
)
s
f
E
(
iu
)
du
(
v
√
N
)
2
−s
f
E
(
iv
)
dv
.
1
/
√
N
u
v
0
This implies that
(
√
N/
2
π
)
s
Γ(
s
)
L
E
(
s
)=
∞
∞
1
/
√
N
(
u
√
N
)
s
f
E
(
iu
)
du
(
v
√
N
)
2
−s
f
E
(
iv
)
dv
−
.
1
/
√
N
u
v
Since
f
(
iu
)
→
0 exponentially as
u →∞
, it follows easily that both integrals
converge and define analytic functions of
s
. Under
s →
2
− s
,therightside,
hence the left side, is multiplied by
−
. This is precisely what the functional
equation claims.
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