Download On the Hurwitz Function for Rational Arguments

On the Hurwitz Function for Rational Arguments
V. S. Adamchik
Department of Computer Science
Carnegie Mellon University
Pittsburgh, USA
[email protected]
Abstract
Using functional properties of the Hurwitz zeta function and symbolic derivatives of the trigonometric functions, the function ζ(2n + 1, p/q) is expressed in
several ways in terms of other mathematical functions and numbers, including in
particular the Glaisher numbers.
2000 Mathematics Subject Classification. Primary 11M35, 33B99. Secondary
11B75, 33E20.
Key Words and Phrases. Riemann zeta function, Hurwitz zeta function, multiple gamma
function, Stirling numbers, Bernoulli numbers, Euler numbers, Glaisher’s numbers,
derivatives of the cotangent.
1
Introduction
The Hurwitz zeta function, one of the fundamental transcendental functions, is traditionally
defined (see [3, 4, 15, 17]) by the series
ζ(s, a) =
∞
X
k=0
1
,
(k + a)s
<(s) > 1,
a∈
/ Z−
0
(1)
The series converges absolutely for <(s) = σ > 1 and the convergence is uniform in every
half-plane σ ≥ 1 + δ (δ > 0). Therefore, ζ(s, a) is analytic function of s in the half-plane
<(s) = σ > 1. It satisfies the following integral representation (see [3], theorem 12.2)
Z ∞ s−1 −ax
x e
1
dx, <(s) > 1, <(a) > 0
(2)
ζ(s, a) =
Γ(s) 0 1 − e−x
Using the Taylor expansion 1/(1 − ex ) = − x1 + 12 + O(x) and splitting (2) into two integrals
µ
¶
Z ∞
1
1
1 1
s−1 −ax
ζ(s, a + 1) =
x e
dx
− +
Γ(s) 0
1 − ex x 2
µ
¶
Z ∞
1
1 1
s−1 −ax
+
x e
−
dx
Γ(s) 0
x 2
we can analytically continue ζ(s, a) into the strip −1 < <(s) < 1. Evaluating the second
integral, replacing a by a + 1 and making use of
ζ(s, a + 1) = ζ(s, a) −
we obtain
a1−s
a−s
1
ζ(s, a) =
+
+
s−1
2
Γ(s)
Z
1
,
as
<(a) > 0
µ
∞
x
s−1 −ax
e
0
(3)
1
1 1
− +
x
1−e
x 2
¶
dx,
(4)
<(s) > −1, <(a) > 0
This could be further generalized to (see [17, p.93])
ζ(s, a) = a
−s
+
n
X
Γ(k + s − 1) Bk
Γ(s)
k=0
1
Γ(s)
Z
∞
Ã
xs−1 e−ax
0
k
a1−s−k +
n
X Bk
1
−
a1−k
1 − ex k=0 k
!
dx,
(5)
n
<(s) > −2b c − 1, <(a) > 0
2
where Bk are the Bernoulli numbers.
We can also analytically continue the Hurwitz zeta function to the whole complex s-plane
(except for a simple pole at s = 1) by means of the contour integral
Z s−1 ax
x e
Γ(1 − s)
dx
(6)
ζ(s, a) =
x
2πi
C 1−e
where the contour C is a loop that starts from −∞ along the lower side of the real axis,
encircle the origin and then returns to −∞ along the upper side of the real axis.
Among all integral representations for ζ(s, a), the Hermite integral has a definite place.
Its derivation is based on the summation formula of Plana (see [4, p.22] or [17, p.90])
Z ∞
Z ∞
∞
X
f (ix) − f (−ix)
1
f (x) dx + i
dx
(7)
f (k) = f (0) +
2
e2πx − 1
0
0
k=0
2
Choosing f (k) in (7) as
f (k) =
1
(k + a)s
after some obvious evaluations, we readily find
Z ∞
a−s
a1−s
(a + ix)−s − (a − ix)−s
ζ(s, a) =
+
+i
dx,
2
s−1
e2πx − 1
0
(8)
<(a) > 0
Noting that
(a + ix)−s − (a − ix)−s =
i (a2
2
x
sin(s arctan( ))
2
s/2
+x )
a
(9)
we rewrite (8) in an equivalent form
a−s
a1−s
ζ(s, a) =
+
+2
2
s−1
Z
∞
0
¡
¡ ¢¢
sin s arctan xa
(x2 + a2 )s/2 (e2πx − 1)
dx,
(10)
s 6= 1, <(a) > 0
This integral is known as Hermite’s integral of ζ(s, z).
The Riemann zeta function ζ(s) is a special case of the Hurwitz zeta function
ζ(s, 1) = ζ(s)
Many series and integral representations for ζ(s) follows straightforwardly from correspondent representations for the Hurwitz zeta function. The same as the latter, the Riemann
zeta function is an analytic function in the whole complex s-plane, except the pole at s = 1.
Another special case of the Hurwitz function is the Bernoulli polynomials
ζ(−n, a) = −
Bn+1 (a)
,
n+1
n ∈ N0
This formula follows directly from integral (6) by means of the residue theorem
µ
¶
Z −n−1 ax
Γ(n + 1)
x
e
1 xeax
ζ(−n, a) =
dx = −n! Res
x
x=0
2πi
xn+2 ex − 1
C 1−e
(11)
(12)
and the generating function for the Bernoulli numbers
∞
X
xn
xeax
=
B
(a)
n
ex − 1 k=0
n!
3
(13)
The Hurwitz function is closely related to the multiple gamma function Γn (z) defined
as a generalization of the classical Euler gamma function Γ(z), by the following recurrencefunctional equation (for references and further reading see [1, 2, 5, 17]):
Γn+1 (z)
,
Γn (z)
Γ1 (z) = Γ(z),
Γn (1) = 1.
Γn+1 (z + 1) =
z ∈ C,
n ∈ N,
(14)
The simplest functional relationship between ζ(t, z) and Γn (z) is given by (we refer the reader
to [1] for other relations of this type)
log Γ2 (z + 1) + z log Γ(z) = ζ 0 (−1, z) − ζ 0 (−1),
where ζ 0 (t, z) =
integral (10):
d
ζ(t, z)
dt
<(z) > 0,
(15)
is defined as an analytic continuation provided by the Hermite
z2
z2 z
ζ (−1, z) =
log z −
− log z +
2
4
2
Z
∞
0
0
2 z arctan( xz ) + x log(x2 + z 2 )
dx,
e2πx − 1
(16)
<(z) > 0.
The constant ζ 0 (−1) = ζ 0 (−1, 0) is known as the Glaisher-Kinkelin constant A, defined by
log A =
1
− ζ 0 (−1).
12
(17)
It satisfies the following integral representation
1 + log(2π)
1
log A =
− 2
12
2π
Z
∞
x log x
dx.
ex − 1
0
(18)
The constant A originally appeared in papers by Kinkelin [11] and Glaisher [7, 8] on the
asymptotic expansion when n → ∞ of the following product
p
p
p
11 22 . . . nn ,
p ∈ N.
For p = 1, we have
µ
1
1 2 . . . n = n! Γ2 (n + 1) = A exp ζ (−1, n + 1) −
12
1
2
n
n
0
Generally,
1p
2p
1 2
...n
np
³
´
= exp ζ (−p, n + 1) − ζ (−p)
0
4
0
¶
(19)
(20)
From the integral (6) one can derive the Hurwitz series representation
Ã
!
∞
∞
−2πina
2πina
X
X
Γ(1 − s) i
e
e
ζ(s, a) =
e−πis/2
− eπis/2
1−s
(2π)1−s
n
n1−s
n=1
n=1
(21)
valid in the half-plane <(s) < 0 and 0 < a < 1. If a 6= 1 this representation is also valid for
<(s) < 1. The Dirichlet series in (21) can be rewritten in terms of the polylogarithm Lim (z):
ζ(s, a) =
¢
Γ(1 − s) i ¡ −πis/2
e
Li1−s (e2πia ) − eπis/2 Li1−s (e−2πia )
1−s
(2π)
(22)
or, equivalently, (for <(s) > 1)
ζ(1 − s, a) =
¢
Γ(s) ¡ −πis/2
−2πia
πis/2
2πia
e
Li
(e
)
+
e
Li
(e
)
s
s
(2π)s
(23)
Setting a = 1, yields the functional equation for the Riemann zeta function
ζ(1 − s) = 2
Γ(s)
πs
cos( )ζ(s)
s
(2π)
2
(24)
The result holds for all admissible s by analytic continuation.
It is easy to see that the polylogarithm Lis (e2πia ) is a linear combination of the Hurwitz
zeta functions when parameter a is rational. Indeed, rearranging terms according to the
residue classes mod q, we obtain
2πip/q
Lis (e
q
q
∞
∞
X
X
X
X
e2πinp/q
1
2πinp/q
) =
=
e
s
(qk + n)
(qk + n)s
n=1 k=0
n=1
k=0
q
1 X 2πinp/q
n
= s
e
ζ(s, )
q n=1
q
Now replacing the polylogarithms in (23) by the above finite sum, we obtain
µ
¶
q
¡
2Γ(s) X
p¢
πs 2πkp ¡ k ¢
=
−
ζ 1 − s,
cos
ζ s, ,
q
(2πq)s k=1
2
q
q
1≤p≤q
(25)
which holds true for all s 6= 0.
The main object of this paper is to derive a family of closed form representations for
ζ(2n + 1, pq ) by evaluating symbolic derivatives of certain trigonometric functions. We will
also show how the results presented here relate to those that were obtained in other works.
5
2
Reflection formula
In this section we express symbolic derivatives of the cotangent function in a closed form.
Simple observation
d
cot(z) = −(1 + cot2 (z)
dz
d2
cot(z) = 2 cot(z)(1 + cot2 (z)
dz 2
n
d
suggests that dz
n cot(z), n ∈ N is a polynomial in cot(z). Generally, some trigonometric and
hyperbolic functions fall to the class of functions whose derivatives are polynomials in terms
of the same function (see [10, 12, 13]).
Lemma 2.1 For any integer n > 1,
µ ¶n
n
X
d
k! n n o
n
cot(z) = (2i) (cot(z) − i)
(i cot(z) − 1)k
k
dz
2 k
j=1
where
©jª
k
are the Stirling subset numbers, defined by [16]
¾ n o
¾ ½
½
nno
n−1
n
n−1
,
= δn
+
=k
k−1
0
k
k
(26)
(27)
Proof. We start with the identity
µ
2
cot(z) = i 1 + 2iz
e −1
¶
Setting eiz → y, we have
µ
µ
µ ¶n
¶n µ
¶
¶n
d
d
2
d
2
eiz →y
n+1
n+1
cot(z) = i
y
1+ 2
=i
y
2
dz
dy
y −1
dy
y −1
Expanding
1
y 2 −1
into a geometric series and then differentiating, we obtain
µ
d
dz
¶n
cot(z)
eiz →y
=
n+1
−(2i)
∞
X
k n y 2k
k=1
Next, we make a use of (see [12])
µ
¶k
∞
1 X n
z
z =
,
k!
k
k
1 + z k=1
1+z
k=1
n
X
nno
k
6
n ∈ N |z| < 1
(28)
which is proved by means of the generating function
½ ¾
n
X
n
n
j
k =
(−1) (−k)j
j
j=1
(29)
where (k)j is the Pochhammer symbol. Substituting (29) into the right hand side of (28),
yields
¶k
½ ¾µ
∞
n
1 XX
z
n
j
(−1) (−k)j
z + 1 k=1 j=1
z+1
j
Changing the order of summation and evaluating the inner sum
µ
¶m
∞
X
m!
y
k
(−k)m y =
1−y y−1
k=1
proves (28).
Therefore, in a view of (28), the derivative of the cotangent immediately reduces to
µ ¶n
n
n n o µ y 2 ¶k
n+1 X
d
eiz →y (2i)
cot(z) =
k!
dz
y 2 − 1 j=1
1 − y2
k
Finally, we set y = eiz into the right hand side to obtain formula (26).
¤
In the similar way we can derive a representation for the derivatives of the tangent
µ ¶n
n
X
k! n n o
d
(i tan(z) − 1)k
(30)
tan(z) = (−2i)n (tan(z) − i)
k
dz
2
k
j=1
To complete the picture, by changing z → iz, we obtain representations for hyperbolic
functions
µ ¶n
n
X
d
(−1)k k! n n o
coth(z) = 2n (coth(z) − 1)
(coth(z) + 1)k
(31)
k
dz
2
k
j=1
µ
d
dz
¶n
tanh(z) = 2n (tanh(z) − 1)
n
X
(−1)k k! n n o
2k
j=1
k
(tanh(z) + 1)k
(32)
Theorem 2.2 For any integer n > 1 and 0 < x < 1,
ζ(n, 1 − x) + (−1)n ζ(n, x) =
n−1
where
©jª
k
¢ X k!
(2πi)n ¡
i cot(πx) + 1
2(n − 1)!
2k
k=1
are the Stirling subset numbers.
7
½
¾
¢k
n−1 ¡
i cot(πx) − 1
k
(33)
Proof. The result follows from the reflection formula for the Hurwitz function
ζ(n, 1 − x) + (−1)n ζ(n, x) = −
π
dn−1
cot(πx),
(n − 1)! dxn−1
n > 1,
0<x<1
and Lemma 2.1.
(34)
¤
Theorem 2.3 For n, p, q ∈ N and 1 ≤ p ≤ q,
p
p
π 2n+1
ζ(2n + 1, 1 − ) − ζ(2n + 1, ) = −
lim
q
q
(2n)! y→0
µ
d
dy
¶2n
sin(2πp/q)
cos(2y) − cos(2πp/q)
(35)
Proof. We replace n by 2n + 1 and x by p/q in formula (33) and then use identity (28) to
obtain
p
p
i(−1)n (2π)2n+1
ζ(2n + 1, 1 − ) − ζ(2n + 1, ) =
q
q
(2n)!
limπip
y→exp(
q
∞
X
)
k 2n y 2k
k=1
where the series is understood as a meromorphic function in y ∈ C
¶2n
µ
∞
X
1
1
d
2n 2k
k y = − 2n y
2
2
dy
y −1
k=1
Hence,
p
p
i(−1)n (2π)2n+1
ζ(2n + 1, 1 − ) − ζ(2n + 1, ) = −
q
q
22n (2n)!
µ
¶2n
1
d
limπip y
2
dy
y −1
y→exp( q )
It is easy to verify by direct evaluation that
¯
µ
¶2n
µ
¶2n
d
1
d
1 ¯¯
y
=− z
dy
y2 − 1
dz
z 2 − 1 ¯z= 1
y
for n = 1, 2, . . . and, therefore,
µ
¶2n
µ
¶2n
d
1
d
1
limπip y
=−
lim πip y
2
2
dy
y −1
dy
y −1
y→exp( q )
y→exp(− q )
Thus, formula (36) transforms to
p
p
i(−1)n π 2n+1
ζ(2n + 1, 1 − ) − ζ(2n + 1, ) = −
·
q
q
(2n)!
Ã
µ
¶2n
d
1
limπip y
2
dy
y −1
y→exp( q )
µ
−
8
lim
y→exp(− πip
)
q
d
y
dy
¶2n
1
2
y −1
!
(36)
Replacing y by y eπip/q and collecting terms, yields
µ
¶2n
2 sin( 2πp
)y 2
p
(−1)n π 2n+1
d
p
q
ζ(2n + 1, 1 − ) − ζ(2n + 1, ) = −
lim y
y→1
q
q
(2n)!
dy
y 4 − 2 cos( 2πp
)y 2 + 1
q
which finally leads to (35) upon substitution y → exp(iy).
3
¤
Special values of ζ(2n + 1, p/q)
In this section we consider special cases of ζ(2n + 1, pq ), pq = 16 , 56 , 41 , 34 , 13 , 23 , 12 when it is expressible in terms of other transcendental functions and constants. Originally this problem
was solved by K.S. Kolbig [14], who expressed ψ (2n) ( pq ) in finite sums of the Bernoulli numbers, using functional properties of the polylogarithm. In our approach, ζ(2n + 1, pq ) are
expressed in terms of generating functions of trigonometric functions.
The case q = 2 is trivial and follows immediately from the multiplication formula (which,
in turns, can be easily proved by rearranging terms in (1) according to the residue classes
mod n)
ζ(s, nz) = n
−s
n−1
X
ζ(s, z +
k=0
k
),
n
n∈N
(37)
Setting z = 1/2 and n = 1 in (37), we obtain
1
ζ(s, ) = (2s − 1) ζ(s)
2
(38)
Next, we will derive other special cases of ζ(2n + 1, p/q) based on the reflection formula (35).
Theorem 3.1 For n ∈ N,
1
ζ(2n + 1, ) )
¡
¢
(−1)n (2π)2n+1
4
= 22n 22n+1 − 1 ζ(2n + 1) ±
E2n
4(2n)!
3
ζ(2n + 1, )
4
(39)
where En are the Euler numbers.
Proof. From the multiplication formula (37) with n = 4, z = 1/4 and Theorem 2.3 with
p = 1, q = 4 we find
¯
µ ¶2n
¯
2n+1
¡
¢
π
1
d
¯
sec(2y)¯
ζ(2n + 1, ) = 22n 22n+1 − 1 ζ(2n + 1) +
¯
4
2(2n)! dy
y=0
9
Recalling the generating function for the Euler numbers
∞
X
zn
sech(z) =
En
n!
k=0
we conclude the proof.
¤
Formula (39) along with (33) suggests a functional relation between the Stirling and
Euler numbers.
Lemma 3.2 For n ∈ N,
(1 − i)
¶k
2n µ
X
i−1
2
k=1
½
k!
2n
k
¾
= E2n
(40)
¶k ½ ¾
2n µ
X
1+i
2n
(i + 1)
−
= E2n
k!
k
2
k=1
where i =
√
(41)
−1.
Proof. We prove the first identity (40). By virtue of (28), we have
(1 − i)
¶k
2n µ
X
i−1
2
k=1
½
k!
2n
k
¾
= −2i lim
z→i
∞
X
k 2n z k
k=1
where the series is understood by the analytic continuation. Furthermore, since
µ
¶n
∞
X
d
z
n k
k z = z
dz
1−z
k=1
we have
(1 − i)
¶k
2n µ
X
i−1
k=1
2
½
k!
2n
k
¾
µ
d
= −2i lim z
z→i
dz
¶2n
z
1−z
It is not difficult to verify the following chain of transformations
µ
µ
µ
¶2n
¶2n
¶2n µ
¶
d
1
d
1
d
z
1
1
= lim z
= lim z
−
lim z
z→i
z→i
dz
1−z
dz
1−z
2 z→i
dz
1−z 1+z
µ
µ
¶2n
¶2n
d
d
z
z
= i lim z
= lim z
2
z→1
z→i
dz
1−z
dz
1 + z2
10
Therefore,
(1 − i)
¶k
2n µ
X
i−1
k=1
2
½
k!
2n
k
¾
µ
d
= lim z
z→1
dz
¶2n µ
2z
1 + z2
¶
(42)
On the other hand,
µ
E2n = lim
z→0
d
dz
¶2n
µ
d
sech(z) = lim y
y→1
dy
¶2n
2y
1 + y2
(43)
where y = ez . Comparing (42) with (43), concludes the proof.
¤
Similarly to Theorem 3.1 we derive closed form representations for q = 3 and q = 6.
Theorem 3.3 For n ∈ N,
1
ζ(2n + 1, ) )
1 2n+1
6
=
(3
− 1)(22n+1 − 1)ζ(2n + 1)
2
5
ζ(2n + 1, )
6
¯
√ µ ¶2n
¯
2n+1
3
d
1
π
¯
±
¯
2(2n)!
dy
2 cos(2y) − 1 ¯
(44)
y=0
1
ζ(2n + 1, ) )
1 2n+1
3
=
(3
− 1)ζ(2n + 1)
2
2
ζ(2n + 1, )
3
¯
√ µ ¶2n
¯
d
1
π 2n+1 3
¯
±
¯
2(2n)!
dy
2 cos(2y) + 1 ¯
(45)
y=0
In particular, setting n = 1 and n = 3 in Theorems 3.1 and 3.3, we obtain
1
1
ζ(3, ) =
ζ(3, ) = π 3 + 28ζ(3)
4
3
√
1
1
ζ(3, ) = 2 3π 3 + 91ζ(3) ζ(5, ) =
6
4
5
2π
1
1
ζ(5, ) = √ + 121ζ(5) ζ(5, ) =
3
6
3 3
11
2π 3
√ + 13ζ(3)
3 3
5π 5
+ 496ζ(5)
3
22π 5
√ + 3751ζ(5)
3
In [9] (see also [6], p. 79), Glaisher defined special numbers Hn and Gn analogous to the
Eulerian numbers
∞
X
3
xn
=
Hn
(46)
2(2 cos(x) − 1) k=0
n!
∞
X Gn xn
3
=
2(2 cos(x) + 1) k=0 n + 1 n!
(47)
Here are the first few values
3
H0 = , H2 = 3, H4 = 33
2
H6 = 903, H8 = 46113, H2n+1 = 0
and
1
G0 = , G2 = 1, G4 = 5
2
G6 = 49, G8 = 809, G2n+1 = 0
It is easy to see that representations (44) and (45) can be rewritten in terms of the Glaisher
constants.
Corollary 3.4 For n ∈ N,
1
ζ(2n + 1, ) )
1 2n+1
(2π)2n+1
6
=
(3
− 1)(22n+1 − 1)ζ(2n + 1) ± √
H2n
2
5
2 3(2n)!
ζ(2n + 1, )
6
1
ζ(2n + 1, ) )
1 2n+1
(2π)2n+1
3
=
(3
− 1)ζ(2n + 1) ± √
G2n
2
2
2 3(2n + 1)!
ζ(2n + 1, )
3
On the other hand, inverting Corollary 3.4, Glaisher’s numbers are just a combination of the
Hurwitz functions:
√
µ
¶
3 (2n)!
1
5
ζ(2n + 1, ) − ζ(2n + 1, ) , n ∈ N
(48)
H2n =
(2π)2n+1
6
6
√
µ
¶
1
2
3 (2n + 1)!
G2n =
ζ(2n + 1, ) − ζ(2n + 1, ) , n ∈ N
(49)
(2π)2n+1
3
3
Acknowledgments
This research was supported by NSF grant CCR-0204003.
12
References
[1] V. S. Adamchik, Multiple Gamma function and its application to computation of series, The
Ramanujan Journal 9(2005), 271-288.
[2] V. S. Adamchik, Symbolic and numeric computations of the Barnes function, Comp. Phys.
Comm. 157(2004), 181-190.
[3] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
[4] H. Bateman, A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York,
Toronto, London, 1953.
[5] J. Choi, H. M. Srivastava, V. S. Adamchik, Multiple Gamma and related functions, Appl.
Math. and Comp. 134 (2003), 515–533.
[6] A. Fletcher, J. C. P. Miller, L. Rosenhead, L. J. Comrie, An Index of Mathematical Tables,
Vol. 1, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962.
[7] J. W. L. Glaisher, On the product 11 , 22 , 33 , . . . , nn , Messenger of Math. 7(1887), 43-47.
[8] J. W. L. Glaisher, On certain numerical products, Messenger of Math. 23(1893), 145-175.
[9] J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London
Math. Soc. 31(1899), 216-235.
[10] M. E. Hoffman, Derivative polynomials for tangent and secant, American Math. Monthly
102(1995), 23-30.
[11] Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung
auf die Integralrechnung, J. Reine Angew. Math. 57(1860), 122–158.
[12] P. M. Knopf, The operator (z d/dz)n and its applications to series, Math. Magazine 76(2003),
364-371.
[13] D. Knuth, T. Buckholtz, Computation of tangent, Euler, and Bernoulli numbers, Math. Comp.
21(1967), 663-688.
[14] K. S. Kolbig, The polygamma function and the derivatives of the cotangent function for
rational arguments, CERN-IT-Reports, CERN-CN-96-005, 1996.
[15] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions
of Mathematical Physics, Springer, Berlin, 1966.
[16] K. N. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, New York,
2000.
[17] H. M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Series
on Mathematics and Its Applications, vol. 531, Kluwer Academic Publishers, Dordrecht, 2001.
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