Internat. J. Math. and Math. Sci.
(1987) 395-403
395
Vol. i0 No. 2
FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES
IBRAHIM A. ABOU-TAIR
Department of Mathematics
Gaza
Islamic University
Gaza- Strip
(Received December I0, 1985 and in revised form May I, 1986)
ABSTCT.
In this paper we study the special Dirichlet series
2
L(s)
sin(---)n
This series converges uniformly in the half-plane Re(s) >
We show that the function
holomorphic function there.
holomorphic function in the whole complex-plane.
the points 0,+/-1,-2,+/-3,-4,+/-5,... are obtained.
L
and thus represents a
can be extended to a
The values of the function
L
at
The values at the positive integers
1,3,5,... are determined by means of a functional equation satisfied by
L.
Dirichlet Series, Analytic Continuation, Functional Equation,
F-Function.
1980 AMS SUBJECT CLASSIFICATION CODE. 30B50, 30B40.
KEY WORDS AND PHRASES.
1.
INTRODUCTION.
By a Dirichlet series we mean a series of the form
n:l
where the coefficients
a
are any given numbers, and
n
s
is a complex variable
[1],
[2].
In this paper we study the special Dirichlet series
L(s)
2=n
_2
n=l
sin(---)n
-s
which converges uniformly in the half-plane Re(s) >
function there.
C
s
and thus represents an analytic
we study the analytic behaviour of the function
In section
L
I, and prove that the function L can be extended to a
beyond the half-plane Re(s)
holomorphic function in the whole complex-plane. Moreover values of L at the points
-m (m=0,I,2,3,...) are obtained at the end of this section. The values of L at the
>
positive integers 1,3,5,... are determined by means of the functional equation
L()
2
(2_)
satisfied by the function L, which we prove in section 2.
I. A. ABOUoTAIR
396
2.
ANALYTIC CONTINUATION OF L.
2___
L(s)
sin
2n n
(---)
-s
(s
is uniformly convergent in the half-plane Re(s)
function there.
LEMMA 2.1.
L
[
2n
t
F (s)
e
N, for
n
-t
-nt
,Re(t)>o
+1
t
r(s)
-t s-I
t
dt
in the above integral yields
#
e
J
>
,where
Euler’s integral
Co,sider the
-s
sin(----)e
n=l
e +e
Substitution of nt,n
g(t)tS-ldt
2
v/
to the whole complex plane
in the half-plane Re(s) >
s
F(s)
G(t)
Thus for Re(s)
L
C.
is holomorphic in
For all values of
L(s)
PROOF.
and so it represents an analytic
The aim of this section is to extend
and to prove that
(2.1)
C)
-nt st
dt
,Re(s)>o
I, we get
1(s)L(s)
__2
/
n=l
/ e-ntts-1
sin.-m--)(2=n_
dt
i.e.
sin
V
n=l
(--}--) e
t
dt
Thus
l(s)L(s)
Now
C(t)
iv
Joe,
G(t)
f
C(t)tS-ldt
n
((g)n-()n)e --nt
n=l
i ([[
()n
i
By using the identities
V/
(1
i/
E
G(t)
e
-nt
()n
e-nt
n=l
Thus
G(t)
e
,where E
-
t +e -t +I
g
+
e- t
+
(1
0
and
e
2rI/3
,Re(t)>O
e- t
i, we get
-
397
FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES
-I
-t
t
is analytic near t=0; therefore it can
+ i)
(e + e
The function G(t)
So we have
be expanded as a power series in t.
G(t)
LEMMA 2.2.
has the Taylor series expantion
G(t)
a t
2n
Itl<
R=O
a
where the coefficients
ao
PROOF.
satisfy the recursion formula
n
n
s
3s n + 2
n-k
n
I/3
G(t)
R=O
an t2n
whi=h i vd na -o (+/-n f= va+/-d
nearest singularities
-+--
t
n
of G(t)).
II
h d+/-k
The relation
<
2
G(t)(e
wh=h x=nd =o h
t
+
e
-t
+ i)
gives
(2n),)=
n
n=o
can be expressed as
G
is an even function, the expantion of
G
Since
(2.2)
i.e.
n t2n
n=o
2(
+
n t2n)(
n=o
_
i.eo
an t 2n
n=o
Thus for the coefficients
n=o
a
n
n=o
t
(2n)’! )=1
t 2n
n-k
we have the recursion formula
n
ao
1/3
3a
n
+ 2
(-E-)-!
a
0
n-k
n>l
k=l
This completes the proof of the lemma.
The coefficient
a
n
easily determined to be
can be determined successively by (2.2).
ao
3
al
9
a2
3T
a3
-1080
THEOREM 2.1.
The function
L(s)
!
-I’(s
L
The first few are
7
defined by
C(t)
ts-1 dt
,Re(s)>1
can be extended to a holomorphic function in the whole complex plane.
PROOF.
Let us define
P
and
Q
for Re(s) >
P(s)
/
G(t) ts-ldt
Q(s)
f
G(t)t s-ldt
by
I.A. ABOU-TAIR
398
The integral
f
G(t)t
s-ldt
exists and converges uniformly in any finite region of the s-plane, since the function
(e -t t Re(s)+l )/(e -t + e -2t +I
is bounded for all values of Re(s), and we can compare the integral with that of
i/t 2.
Q
Thus
Recall from Lemma 2.2 that
is an entire function.
g(t)
=--
2n
a t
,t
n
n=o
We deduce for Re(s)
the convergence being uniform on [0,i].
=’
P(s)
a
n=o
n
,[ o,1]
that
t2n+s-ldt
n=o
P
Thus
I/F
Since
Q
Since
P/F
__P(s)
and
I/F
P
(i)
L
can only be those
L
Since
is regular for all values of
(ii)
The values of
L
at 0,-2,-4,-6
0 ,1,2,5,4
,m
(2m)!a
This follows immediately from the fact that
P(1-2m)
F(1-2m)
t(1-2m)
Q(1-2m) =0
+
F(1-2m)
As in (i) we use the partial fraction (2.3) of
L(-2m)
P(s) + (s)
F( s
lim
s-9-2m F( s
lim
s--- 2m
P(s)
F(s)
s
-,-l’im2m
i.e.
L(-2m)
s
are given by
I/r
has zeros at 0,-I,
-2,-3,..., and thus
(li)
I/F
has zeros at -I,-3,-5
L(-2m)
(i)
L
has simple poles at 0,-2,-4,-6
This completes the proof of the theorem.
in the complex plane.
LEMMA 2.3.
by
(2.3)
are entire functions, the singularities of
We have seen that
C
to the whole of
Q(s)
r(s)
+
r(s)
has simple zeros at 0,-2,-4,... it follows that
PROOF.
L
is an entire function we may now extend
L(s)
of
with simple poles at 0,-2,-4,-6
C
is a meromorphic function on
lim
s-->-2m
r s)
2m/s
a
m
L
,m,N
-
to get
__1__1
[’(s) z___n=o
2n+an
FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES
F
Since
has simple poles at the points
399
(m=0,I,2,3,...) with residues
-m
(-l)m/m!,
we get
lim
(2m+s) F (s)
s -9 2m
Res(r ,-2m)
(2)’---"
m
Thus
L(-2m)
where
3.
a
(2m)!a m
,m
0,1,2,3,...
can be determined successively by (2.2).
m
DERIVATION OF THE FUNCTIONAL EQUATION OF L.
In this section we derive the equation
2___( r(-)o(/-)L(_)
L(s)
.,/Y
where
seres
is the Dirichlet
2
L(s)
(2.1)
L
Finally we determine the values of
There exists an integral function
L(s) =--P(1-s)I(s)
PROOF.
C
2
c
C
and
s
0 < r < i, and let
Let
C
at 1,3,5,..., by the use of the functional
equation obtained above.
LEMMA 3.1.
-
sin( 2n )n -s
4T n=
,s
Cr
I
such that
C
be the contour consisting of the paths
C I,
C 3, where
((R),r]
2
+Dr(0)
is a circle of radius
r
and the center at the origin oriented in the
positive direction.
C
,,c.
3
[r,).
C
I
Define the function
r
I r(s)
We prove now that
I
r
Ir(S)
by
I_/_
2i
f
C
r
(-t) s-1
-t
e
et+
We have
is independent of r.
It’(s)
-Y
dt
+
f
Co
(_t) s-1
t
-t
e + e
+1
I.A. ABOU-TAIR
400
where
C
dE
et -t+
Co
where
(a).
is the contour shown in figure
O
Now
im
8-->o
e +
et -t+1
C
-dt
e +
is the contour in figure (b).
C
According to Cauchy’s theorem, the integral around
(-t)sC
t
e +e
o
It follows that
I
is independent of
r
is zero.
Thus
0
dt
-t +1
C
r.
Now,
r
I r (s)
2i
C
2=i
t
t+l
e +e
2
e
|
(log t -=i)(s-1)
t -t
e +e +1
dt+
t
-t
e +e
+1
r
provided Re(s) > O, since
0
r
+
dt
J
The middle term approaches zero as
dt
2
C
t
2
e +e
< M
dt
-t +1
e
r
< H
r
Re(s)
Hence
lim
r--,o
Define the function
l(s)
_e
I r (s)
I
-=i(s-1
+e
=i(s-1
2,i
y
t s-1 ---dt.
e +e
+1
by
lim
r--o
I r (s)
Thus we have
l(s)
=--sin(ms)
ts-1
i et+e-t+l
dr.
We have seen in the proof of theorem 2.1 that the function defined by the integral
ts_i
t -t
e +e +1
is a meromorphic function with simple poles at the points 0,-2,-4,
function sin(s) has simple zeros at 0,-2,-4
it follows that
Since the
I
is regular for
FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES
all values of
401
in the complex plane.
s
Moreover we have
_[(s)sin(s)
l(s)
t(s)
Thus
L(s)
I(s) r(1-s)
THEOREM 3.1.
L
The function
s-1
2
#()
L(s)
satisfies the functional equation
r(-s)cos(
s)t(-s)
n+, n 1,2,3
and let C
(0<r<l) be the contour
n
n,r
to r, a circle radius r and center
consisting of the positive real axis from R
n
at the origin oriented in the positive direction, the positive real axis from r to
PROOF.
R
n’
R
Let
R
and finally a circle of radius
with center at the origin oriented in the
n
negative direction.
i.e.
n,r
r
To deduce the functional equation of
f
---q2
If we assume
C
s
L
we evaluate the integral
(_t) s-1
-dt
et+e_t +I
r,n
Rn
n
is a negative real number, then we have
x
e(X-1)lg(-t)
(-t)x-1
It follows that
x-1
x-1
I(-t)i
Since the function
Itl
(et+e-t+l) -I
is bounded on the circle
_D R
(0),
n
DR
which goes to zero as
n
Now between
I)
(0)
R
lim
n--
2=i
-5--
n
(-t)s-1
C
and
D (0)
-+
--5-(3m+I)
r
n
+/-
-
2 M
goes to infinity.
Thus we have
l(s)
<
t
t
e +e -t
(o)
2=i
t
e +e
n,r
-t
dt
+1
the integrand has poles at the points
and
+/-
(3m-I), m=1,2,3,
I. A. ABOU-TAIR
402
Denote
(_t) s-1
H(t)
t +e -t +I
e
Thus we have
2i
Res (H, ----)
2=
1__( ----)
e -=is/2
V/
2i
) --x/ (2:)
Res(H,-
s-1
Res(H,2---{m+l))
s-1
,is/2
--(2,)
2,i
2,
e
-(-2,)
Res(H,-----(3m+l))
i
Res(H,----()m-1
s-1
S-1
e
s-1
2
-,is/2
=--()
e
s-1
(3m+I)
,is/2
-is/2
s-1
(3m+l)
s-1
(.,,)m-1)
2-i
2
Res(H,----(3m-1)
=---(- )s-1 e’iS/2(Sm_l)
D
The sum of the residues between
(0)
R
9Dr(0)
and
n
(’2,)
n
s-1
[(3m+1)
cosC,s) (I+Z
m=l
s-1
equals
s-1
-(3m-1)
s-1
)
One can easily verify the identity
n
1+
[()m+l)
m=
)
s-
s-
]
-(3m-1)
sin(,m)m
-1
Thus the sum of the residues is
s-1
__2 (z
2
cos(gs)
m=l
Z
sin(-m) mS-1
)
It follows that
2
sin(-m) mS-1
2
--(-2) cos(1/2zs) (
s-1
__(22=) cos({xs)L(1-s)
-[(s)
We have seen that
-I(s)F(l-s)
L(s)
the formula (3.1) is true for all
s
(3. i)
s E C, so by the identity theorem
for all
E
).
Thus we have proved the functional
C.
equation
t(s)
LEMMA 3.2.
2
2
s-1
-()
The values of
cos(y,s)
L
(1-s)L(1-s)
s=2m+l,m=O,1,2,3,...
at the points
the formula
L(l+2m)
where
a’s
m
(-1)m
vr2,.(
are determined by (2.2).
a
are given by
FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES
PROOF.
For
s
L(-2m)
-2m
403
the functional equation and the identity
(2m)! a m
m
D,I,2,...
of the previous section give the proof of the lemma.
REFERENCES
I.
2.
HARDY, G.H. and RIESZ, M. "The General Theory of Dirichlet Series," Cambridge
University Press, 1952.
TITCHMARSH, E.C. "The Theory of the Riemann Zeta-Function," Oxford University
Press, 1951.