On analytic continuation and functional equation of certain Dirichlet series

(1)

VOL. 16 NO. 3 (1993) 609-614

RESEARCH

NOTES

ON ANALYTIC

CONTINUATION

AND FUNCTIONAL

EQUATION

OF

CERTAIN OIRICHLET SERIES

E. CARLETTIandG. MONTIBRAGADIN Dipartimentodi Matematica

Universit/tdiGenova ViaL.B. Alberti 4

16132

Genova,

Italy

(Received July 16, 1990 and in revised form December I0, 1992)

ABSTRACT.

Analyticcontinuationandfunctionalequation ofRiemann’stype areproved for a

class of Dirichletseriesassociated to rational functions.

KEY WORDS AND PHRASES.

Dirichlet series, analytic continuation, functional equation,

Hurwitz zeta function.

1991AMS

SUBJECT

CLASSIFICATION CODES.

30B50,30B40.

1. INTRODUCTION.

In

this paper we are concerned with analytic continuation and functional equation

(of

Riemann’s type) ofDirichlet series. Ourobjectiveis toshow howavery classicalmethod, which is oneof Rim’ methods

(see

[11),

works foraquite general class of Dirichlet series.

In

[2]

thismethodisapplied toDirichletL-series L(s,x)where

x

is thenon-principalDirichlet character

rood 3.

In

thecourseof the proofit turns out that theDirichlet series weconsiderinourpaperare

expressedinterms ofHurwitzzeta functions. Thisresult shouldbecompared withTheorem of Arakawa

[3]

wherearepresentation of E cotgamrn in terms of

Barnes

zeta functionsisgiven.

Let

L(s)=,E=

tann

-s _be _a _{Dirichlet series with}

anC, having finite abscissa of absolute

We

assume that the power

series,,anz

n,=

associated to /(s), defines a

(non

convergence O"_o

constant)

rationalfunction G(z)=P(’)such that"

(1)

OP_<=

(2)

(z) has zeros _rl,-..,rm, where

rj=ez(2i#j),#j[-1/2,

1/2)for

l<j<m with

multiplicity aj

(3)

PO’j)

#0for _<j<m.

From

the above assumptions it follows that the radius of convergence p of

n=an

zn

is 1. We provethe following:

TIIEOKEM. LetL(s) and G(z) befunctions withthe abovehypotheses. Thenwehave

(1)

ifz isaregularpoint for G(z), then L(s)canbecontinued toanentirefunction;

(2)

ifz is apolefor G(z)of order n_o>1, then L(s) canbe continued to a meromorphic

functionover Cwithasimplepoleat s n_oandpossibly simplepolesats 1,2,

.,

_no 1.

(2)

(s)

y

Oj>O

(nj-l)!h=OE

(2,)

()

h!T)

h-

[(-i)

s-h(s+h,l_Oj)+

nj-I

E

+

Oj<O

(nj-l)!

h=0 no

+i-s-hg(s+h,l+Oj)]+(no_i)

h=0

where’he

last sum appears ifQ(I)=0 with multiplicity n_o, g(s,a) is the Hurwitz zeta function with 0<a< and

T

are suitable constantscomputed in 53, then ,I,(,) is an entire functionand the function

satisfiesthefunctional equation (s)= (I- s).

Wenote that the class ofDirichletserieswhichsatisfyourhypothesescontains strictlythat

of linear combinationofshifted DirichletL-seriesL(s-k,X),knon-negativeinteger. 2.

SOME LEMMAS.

LEMMA

2.1.

Let

G(z)=

lan

zn

be a complex power series with radius of convergence

p,=

1. Ifz isa pole ofordern_o_> for G(z), thenthe Dirich]et series L()=

.=Elan

n has a

meromorphic continuation over

c

withsimple polesat s n_oandpossiblyat s _1,...,%-1. If z isaregularpoint, then L(s)iscontinuableasanentirefunction.

PROOF. From

the classicalintegralrepresentation of r(,)onegets

.’’--Jlan[

-nt

ts-1

(2.1)

L(s)r(s) dt

0

for Re sufficientlylarge.

Thefunction G(e-

t)

is0(e-

Kt)

forasuitable K >0, as t-

+

oo, andisinfinite of order%, as t-,0

+,

providedz is apole. Sowegetfrom

(2.1)

I

G(e-t)

tS- dt (large).

(2.2)

L(,)r(,) G(e-

t)

t"-

dt+

Thelastintegralis an entirefunctionwhich will bedenoted byM(,).

Let

G(e-t)=

ant

n bethe

Laurent

expansionat 0; we can suppose thatitsouter radius Aisgreater than (forotherwisewewrite

A

instead in(2.2)). Then

I

1G(e-’)

‘’-l

d‘=

I

an

tn+s-l

d’=

n--+-’"

"

(2.3)

0 0n= -n_o n= -n_o

Thelastseriesin

(2.3)

definesameromorphicfunction with simplepolesats n_oandpossiblyat

s n<no, sinceit converges uniformly onany compact subset of {

e

C:

Is

+

n[ >_C,n>

no},

C

beingfixed positive constant.

From

(2.2)

and

(2.3)

onehas

L(,)r(,) M(,)

+

.

+,,

--FI_O

sothatourfirstclaimfollows, becauseof thepolesofr(s)at non-positiveintegers.

(3)

LEMMA

2.2. Let L(s) and G(:)satisfy the hypothesesoftheTheorem. Let usconsider the

entirefltnction

then for a_>

l(s)

_1

r(s) F(s)sin

M

+

o

I(1 s)

Z

E

Res(II(z)( z)-s_,zj,

l),

j- k- cx

where H(z)=G(e-z) _{and zj,}_k

=2ri(-Oj+k)

are the non zero complex numbers such that

exp( _zj,

k)

_rj, <j

<.

m.

PROOF. Let

CN,

and

r

_rbe thecontoursdrawn below. Weput

CN,

r

Here (-z)s- isdefined tobeezp((s- 1)log(-z)), where the logarithmisreal onthe positive real

axis. If is sufficiently small and N#

I’-’j,

_k for eachj,k, then theabove functionsareanalytic

functions of s, because they are expressed as sums of absolutely convergent integrals with

parameter of univalentholomorphicfunctions.

N,r

l-r.

We know that, byastandard argument basedon Cauchy’stheorem, lr(s isindependent on r.

Moreover,

sinceIt(z)(- z)

no

isholomorphic,onegets

I

Il

--rH(z)(-z)S-1

dz[

<2rLe[tlra-n

whereL sup H(z)( z)

n I:

z _{r} and}s a

+

it. Itfollowsthat,fora>no,

Hence,

for#>_no

lira

f

H(z)(- z)s- dz O. r--,O

l(s)

l

L(s)r(s)sinrs

ly

sinrs

H(t)ts-

dt 0

_1

Io

H(t)ezp((s- 1)(logt- ri))dt-

[c

J0 H(t)ezp((s- 1)(Iogt+ri))dt

lim

[

r-0

-

J

Fr

H(z)( z)

s- _dz

(4)

The above equality holdsonthewholecomplexplaue for analytic continuation.

Now,

we can fix some 6>0 and choose N such that

N-Izj,

_k >_ in order to ol)tain H(:)I _<K, with K g(6), for

zl

N. Weobtain, from

If

’l

=N

H(’)(-z)S-1

d:

<-2"Keltl

No.

that, foro.<0

Finally foro.>

lira

f

H(z)( z)s- dz O.

J

Izl

:IV

M +oo

l(1 s) limlr(1 s) lira

IN,r(1

s)=

E

Res(H(z)(-z)-S’zj,

k)"

j= k= -o

3. PROOF OF

THE

THEOREM.

Itisclear that

Lemma

2.1gives immediately the analytic(meromorphic)continuation. Itremains to prove functionalequation.

An

easycomputation shows

nj-1

Res(H(z)(_z)-S_zj,

k)=

-h-s(-(n

j-l)’

E

(2,)

)h!

T(i(-0j+k))

s

’

h=0

where

T=(-l)h(

nj-1

s(nj-l-h)(zj,

k)

with S(z)=H(z)(z-

zJ, k)

nj Wenotethat

T

doesnot dependont.

Fromthedefinitionofl(s)and well-known properties of r(s)itfollows that

L(1 ,) ( or(s). (3.1)

For

o.> we canapply

Lemma

2.2to

(3.1)

sothatweget

J)

nj-I

L(I s) r(s) h s

Z

i-s-h

(Oj-k)-s-h

Ik_

O(

i) s

h(k

tj)

s h

+

<0

-’-

h(-)-’-

a]

+

no

,=oZ

n

j-1

F(s)(2:r) s

E

,,.1

Z

(2r)-h(-)h’T[i-s-h((s+h’Oj)+(-i)-s-h

Oj

>O

(hi-l)!

h O

n

j-1

(5)

\!

(

C(s+h,l+Oj)]+(no_l)

!

]]

(2) l,

-i, h!To

c(s+h)[(-i)

-s-j’+i-

-hi

h=0

(2r)-The function (s) is entire becauseof well-known properties ofHurwitz and Riemann zeta functions. Sothe equality

L(I s) (27r)-SF(s)b(s)

holds, by analytic continuation,onthewhole colnplexplane. Inparticular

(3.2)

hence

L(s) (2)

it(1

s)(1 s),

{(s) (2r)-

*r(s)(s)L(,).

(which is meromorphic on

C)

is invariant by s-.1-s. Wenote that the set of (simple) polesof

{(s) is contained in n_o

+

1, ,no}ifn_o> andit isempty ifn_o O.

4. EXAMPLES.

The hypotheses of the Theorem arefulfilled by those Dirichlet series whose coefficients are

polynomials having rational generatingfunction. For instance, let Tn(z (resp.

Vn(z),CPn(Z))

be Tchebychev polynomials of the first kind (resp. of the second kind, Gegenbauer polynomials),

i.e., for

I

<

(see

[4])

2z2

+

2zz

E

Tn( )zn

n>

z2-2z+

9

(1-

a’2)(-

z"+2zz)

Un(x)z

n> z"-2zz+l

CPn(z)zn=

ppositive integer.

n>0 (z2-2xz+

I)P

The correspondingDirichlet series

E

Tn(z)

n-S’

E

Un(z)n-S’

E

CnP(Z)n-S

(4.1)

n>l n>l n>l

have analyticcontinuationandfunctionalequation.

Let

x

beaDirichlet characterrood q

(not

necessarilyprimitive). Then L(,X)= X(n)n

-

is

q

associated to the rationalfunction G(z)=P(*) where Q(z)=zq and P(z)=

-

(n)z

n.

If is

not principal then P(1)=0 andsoG(z)must beexpressedasquotient oftwocoprime polynomials

by dividingP(z) andQ(z)byz-1.

In

thiscasethefunction4(,) takesaverysimple form

with

[q/2]

(’)=

E

[Aj(s)((s,)+Ij(s)((s,I-6)]

j=l

Aj(s)=Re(RJj cosrs/2)-lm(R-Jj

sinrs/2),

Itj(s)=Re(R--Jj

cosrs/2)+Irn(R-Jj

sinrs/2)

(P(z)

where

Rj

Res

ikQ(z),rj

(6)

x is primitive and real, whereas it takes an apparently different form if x is primitive and complex. IndeeditconnectsL(1-8,)andL(8,x) insteadofL(8.).

Wepointout that if

arg(x+i(l-z2))

is not arational multipleof=,eachofseriesin

(4.1)

is notalinear combinationofshiftedDirichletL-series.

REFEREN(TE$

1.

TITCHMARSH,

E.C., The Thcory of the Riemann Zeta Function, Oxfi)rd Univ. Press

(second

edition)

(1988I.

2.

ABOU-TAIR,

I.,

Functional equation ofa special Dirichlet series,

Internat.

J. Math. and Math. Sci. 10(2) (1987),395-403.

-s _Dedekind

sums, and Hecke L-functions 3.

ARAKAWA,

T., Dirichlet series cot.q,,,,,

for real quadratic fields,Comm. Math.Univ. Sancti Pauli37

(2)

(1988),209-235.

4.

MAGNUS,

W.

&

OBERHETTINGER, F.,

Formeln und Satze fur die speziellen,