Remarks on the mixed joint universality for a class of zeta-functions

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by  Roma Kacinskaite

mjfi-10.tex August8,2016
REMARKS ON THE MIXED JOINT UNIVERSALITY
FOR A CLASS OF ZETA-FUNCTIONS
ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
ABSTRACT. TworemarksrelatedwiththemixedjointuniversalityforapolynomialEu-
lerproductj (s)andaperiodicHurwitzzeta-functionz (s,a ;B), whena isatranscen-
dentalparameter,aregiven.Oneisthemixedjointfunctionalindependence,andtheother
6
isageneralizeduniversality,whichincludesseveralperiodicHurwitzzeta-functions.
1
0
Keywords: functional independence, periodic Hurwitz zeta-function, polynomial Euler pro-
2
g duct,universality.
u
AMSclassification: 11M41,11M35.
A

5
1. INTRODUCTION

]
T In our former paper [5], we have shown a certain mixed joint universality theorem,
N
which is valid for an Euler product of rather general form, and a periodic Hurwitz zeta-
.
h function.
t
a
Inthepresentnote,wegivetworemarks relatedwiththeresultin[5]. Thefirstremark
m
is on the mixed joint functional independence result. It is well known that functional
[

independence properties can be deduced from universality results. We will show that
2
v such afunctionalindependenceis alsovalidinourpresent mixedjointsituation.
5
9 Thesecondremarkisonageneralizationoftheresultin[5]. Itisanimportantproblem
7
tostudyhowgeneralthemixedjointuniversalitypropertyholds. Wewillproveagenera-
3
0
lizedlimittheoremandageneralizeduniversalitytheorem,whichinvolveseveralperiodic
.
1
Hurwitzzeta-functions,underacertain matrixcondition.
0
6
1
2. FUNCTIONAL INDEPENDENCE
:
v
i The history of the problem on the functional independence of Dirichlet series goes
X
r back to the famous lecture of D. Hilbert [2] in 1900. He mentioned that the Riemann
a
zeta-function z (s)does notsatisfyany non-trivialalgebraicdifferentialequation.
Recall thedefinitionofz (s). LetPbethesetofallprimenumbers,andCthesetofall
complexnumbers. Fors=s +it ∈C, z (s)isgivenby
¥ −1
z (s)= (cid:229) 1 = (cid:213) 1− 1
ms (cid:18) ps(cid:19)
m=1 p∈P
for s >1, and can be analytically continued to the whole complex plane C except for a
simplepoleat thepoints=1 withresidue1.
1
2 ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
In 1973, S.M. Voronin proved [16] the following functional independence result on
z (s). LetNandN bethesetofpositiveintegers,andnon-negativeintegers,respectively.
0
Theorem A ([16]). Let N ∈N andn∈N . Thefunctionz (s)does notsatisfyanydiffer-
0
entialequation
n
(cid:229) sjF z (s),z ′(s),...,z (N−1)(s) ≡0
j
j=0 (cid:16) (cid:17)
forcontinuousfunctionsF , j=0,...,n,notall identicallyzero.
j
Laterthis resultwas generalized to otherzeta- and L-functions. Forthesurvey,see the
monographsby A. Laurincˇikas[6]and J.Steuding [15].
Nowadaysinanalyticnumbertheory,theinvestigationofstatisticalproperties(andalso
thefunctionalindependence)foracollectionofvariouszeta-functions,someofthemhave
the Euler product expansion and others do not have, is a very interesting problem since
an importantroleisplayed by parametersincluded inthedefinitionofthosefunctions.
The first result in this direction is due to H. Mishou. In 2007, he proved [12, Theo-
rem 4] that the pair of zeta-functions consisting of the Riemann zeta-function z (s) and
theHurwitzzeta-function z (s,a )isfunctionallyindependent.
We recall that the Hurwitz zeta-function z (s,a ) with a fixed parameter a , 0<a ≤1,
isdefined bytheDirichletseries
¥
z (s,a )= (cid:229) 1
(m+a )s
m=0
for s > 1, and can be continued to the whole complex plane, except, for a simple pole
at the point s = 1 with residue 1. In general, the function z (s,a ) has no Euler product
overprimes,exceptforthecasesa =1anda = 1,whenz (s,a )isessentiallyreducedto
2
z (s). ThenMishou’sresultis thefollowingstatement.
Theorem B([12]). Supposethata istranscendental. LetN ∈N,n∈N ,andF :C2N →
0 j
C beacontinuousfunctionforeach j=0,...,n. Suppose
n
(cid:229) sj·F z (s),z ′(s),...,z (N−1)(s),z (s,a ),z ′(s,a ),...,z (N−1)(s,a ) ≡0.
j
j=0 (cid:0) (cid:1)
Then F ≡0 for j=0,...,n.
j
This is the first “mixed joint” functional independence theorem. Later this result was
generalized to the collection of a periodic zeta-function and a periodic Hurwitz zeta-
functionbythefirst-named authorand A. Laurincˇikasin[4].
Inthispaper,wewillproverathergeneralresultonthemixedjointfunctionalindepen-
dence for a class of zeta-functions, consisting of the so-called Matsumoto zeta-functions
and periodicHurwitzzeta-functions.
Let B = {b : m ∈ N } be a periodic sequence of complex numbers (not all zero)
m 0
with minimal period k∈N, and 0<a ≤1. In 2006, A. Javtokas and A. Laurincˇikas [3]
REMARKSON THEMIXEDJOINTUNIVERSALITY 3
introduced the periodic Hurwitz zeta-function z (s,a ;B). For s > 1, it is given by the
series
¥
z (s,a ;B)= (cid:229) bm .
(m+a )s
m=0
It isknownthat
z (s,a ;B)= 1 k(cid:229)−1b z s,l+a , s >1.
ks l (cid:18) k (cid:19)
l=0
Therefore the function z (s,a ;B) can be analytically continued to the whole complex
plane, exceptforapossiblesimplepoleat thepoints=1withresidue
1k(cid:229)−1
b:= b .
l
k
l=0
Ifb=0, thecorrespondingperiodicHurwitzzeta-functionis an entirefunction.
ThefunctionalindependenceofperiodicHurwitzzeta-functionswasprovedbyA.Lau-
rincˇikas in[7, Theorem1].
Nowwe recall thedefinitionofthepolynomialEulerproductsj (s),or, so-called Mat-
sumotozeta-functions.
e
Let, for m ∈ N, g(m) ∈ N, and, for j ∈ N and 1 ≤ j ≤ g(m), f(j,m) ∈ N. Denote
by p the mth prime number, and a(j) ∈ C. The zeta-function j was introduced by the
m m
second-named authorin[11], and itis defined bythepolynomialEulerproduct
e
¥ g(m)
j (s)= (cid:213) (cid:213) 1−a(j)p−sf(j,m) −1. (1)
m m
m=1 j=1(cid:16) (cid:17)
e
Supposethat
a (j) b
g(m)≤C p and |a |≤ p (2)
1 m m m
with a positive constant C and non-negative constants a and b . In view of (2), the
1
function j (s) converges absolutely for s >a +b +1 (see Appendix), and hence in this
regionitcan beexpressedas theDirichletseries
e
¥
j (s)= (cid:229) ck, (3)
ks
k=1e
e
withcoefficients c . Forbrevity,denotetheshiftedversionofthefunctionj (s)by
k
¥ ¥
e j (s)=j (s+a +b )= (cid:229) ck = (cid:229) ck e (4)
ks+a +b ks
k=1 e k=1
e
with c = k−a −b c . Then, for s > 1, the series on the right-hand side of (4) converges
k k
absolutely.
e
The aim of this paper is to obtain a mixed joint functional independence of the collec-
tion of zeta-functions consisting of the Matsumoto zeta-function j (s) belonging to the
SteudingsubclassS, defined below,and periodicHurwitzzeta-functionsz (s,a ;B).
Wesaythatthefunctionj (s)belongstotheclassSiffollowingconditionsaresatisfied:
e
e
4 ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
(i) thereexistsaDirichletseriesexpansion
¥
j (s)= (cid:229) a(m)
ms
m=1
witha(m)=O(me )foreverye >0;
(ii) there existss j <1 such that j (s) can be meromorphicallycontinued to thehalf-
planes >s j ;
(iii) thereexistsaconstantc≥0such that
j (s +it)=O(|t|c+e )
foreveryfixed s >s j and e >0;
(iv) thereexiststheEulerproductexpansionoverprimenumbers,i.e.,
j (s)= (cid:213) (cid:213) l 1−aj(p) −1;
(cid:18) ps (cid:19)
p∈Pj=1
(v) thereexistsaconstantk >0 such that
lim 1 (cid:229) |a(p)|2=k ,
x→¥ p (x)
p≤x
wherep (x) denotesthenumberofprimes p, p≤x.
This class was introduced by J. Steuding in [15], and is a subclass of the class of
Matsumotozeta-functions. Forj ∈S, let s ∗ betheinfimumofall s forwhich
1
1 T |j e(s +it)|2dt ∼ (cid:229)¥ |a(m)|2
2T Z−T m=1 m2s
holdsforanys ≥s . Then itisknownthat 1 ≤s ∗ <1 (see[15, Theorem2.4]).
1 2
Westatethefirst mainresultofthispaperin thefollowingtheorem.
Theorem 1. Suppose that a is a transcendental number, and the function j (s) belongs
to the class S. Let N ∈ N, n ∈ N , and the function F : C2N → C be continuous for
0 j
j=0,1,...,n. If
e
n
(cid:229) sj·F j (s),j ′(s),...,j (N−1)(s),z (s,a ;B),z ′(s,a ;B),...,z (N−1)(s,a ;B)
j
j=0 (cid:0) (cid:1)
isidenticallyzero,thenF ≡0 for j=0,1,...,n.
j
3. PROOF OF THEOREM 1
To the proof of the mixed joint functional independence for the functions j (s) and
z (s,a ;B), we need two further propositions: the mixed joint universality theorem and
theso-calleddenseness lemma.
REMARKSON THEMIXEDJOINTUNIVERSALITY 5
3.1. Mixedjointuniversalityofthefunctionsj (s)andz (s,a ;B). TheproofofTheo-
rem 1 is based on themixed joint universalitytheorem in the Voronin sense for the func-
tions j (s) and z (s,a ;B). It was obtained by the authors in [5, Theorem 2.2]. We will
givethe statement of this universalitytheorem as a lemma. Let R be the set of real num-
bers, and D(a,b)={s∈C:a <s <b} for any a<b. For any compact subset K ⊂C,
denotebyHc(K)thesetofall C-valuedcontinuousfunctionsdefined on K,holomorphic
in the interior of K. By Hc(K) we mean the subset of Hc(K) consisting of all elements
0
which arenon-vanishingonK.
Lemma 2 ([5]). Suppose that j ∈ S, and a is a transcendental number. Let K be a
1
compact subset of D(s ∗,1), and K be a compact subset of D(1,1), both with connected
2 e 2
complements. Supposethat f ∈Hc(K )and f ∈Hc(K ). Then, foreverye >0,
1 0 1 2 2
1
liminf m t ∈[0,T]: sup|j (s+it )− f (s)|<e ,
1
T→¥ T (cid:26) s∈K1
sup|z (s+it ,a ;B)− f (s)|<e >0,
2
(cid:27)
s∈K2
where m {A} denotes theLebesgue measureof themeasurablesetA⊂R.
Note that for the proof of Lemma 2 we use the joint mixed limit theorem in the sense
of weakly convergent probability measures for the Matsumoto zeta-functions j (s) and
the periodic Hurwitz zeta-function z (s,a ;B). It was proved by the authors in [5, Theo-
rem 2.1].
3.2. A denseness lemma. FortheproofofTheorem1, weneed adenseness lemma.
Definethemapu:R→C2N bytheformula
u(t) = j (s +it),j ′(s +it),...,j (N−1)(s +it),
(cid:0) z (s +it,a ;B),z ′(s +it,a ;B),...,z (N−1)(s +it,a ;B)
(cid:1)
with 1 <s <1.
2
Lemma 3. Supposethata istranscendental. Then theimageofR byu isdenseinC2N.
Proof. Wewillgiveasketch,sincetheprooffollowsinthesamewayas Lemma13from
[4](see, also, [12,Theorem 3]).
Wecan find asequence{t m :t m ∈R}, limm→¥ t m =¥ ,such thattheinequalities
e
|j (j)(s +it )−s |<
m 1j
2N
and
e
|z (j)(s +it ,a ;B)−s |<
m 2j
2N
6 ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
hold for every e > 0 and arbitrary complex numbers s , l = 1,2, j = 0,...,N−1. To
lj
showthis,we considerthepolynomial
s ·sN−1 s ·sN−2 s
l,N−1 l,N−2 l0
p (s)= + +...+ , l =1,2.
lN
(N−1)! (N−2)! 0!
Then, for j=0,...,N−1 and l =1,2,wehave
(j)
p (0)=s .
lN lj
Now,inviewofLemma2andrepeatinganalogousargumentsasintheproofofLemma13
from[4],wecanprovetheexistenceoftheabovesequence{t }andobtainthattheimage
m
ofR bythemapu isdensein C2N. (cid:3)
3.3. Proof of Theorem 1. Now we are ready to complete the proof of Theorem 1. The
essentialideaisduetoVoronin (see, forexample,[17]). Wefirst provethatF ≡0.
n
Supposethat F 6≡0. It followsthat thereexistsapoint
n
a=(s ,s ,...,s ,s ,s ,...,s )∈C2N
10 11 1,N−1 20 21 2,N−1
such that F (a)6= 0. From the continuity of the function F , we find a bounded domain
n n
G⊂C2N such thata∈G,and, forall s∈G,
|F (s)|≥c>0. (5)
n
By Lemma3, thereexistsa sequence{t m :t m ∈R},limm→¥ t m =¥ , suchthat
j (s +it),j ′(s +it),...,j (N−1)(s +it),
(cid:0) z (s +it,a ;B),z ′(s +it,a ;B),...,z (N−1)(s +it,a ;B) ∈G.
(cid:1)
Butthistogetherwith(5)contradictstothehypothesisofthetheoremift issufficiently
m
large. Hence, F ≡0.
n
Similarlywe can showF ≡0,...,F ≡0, inductively. Theproofiscomplete.
n−1 0
4. A GENERALIZATION
Themixedjointuniversalityand themixedjoint functionalindependencetheorem can
beobtainedinthefollowingmoregeneral situation.
Suppose that a be a real number with 0 < a < 1, l(j) ∈ N, j = 1,...,r, and l =
j j
l(1)+...+l(r). Foreach j and l, 1≤ j≤r,1≤l ≤l(j),letB ={b ∈C:m∈N }
jl mjl 0
be a periodic sequence of complex numbers (not all zero) with the minimal period k ,
jl
and let z (s,a ;B ) be the corresponding periodic Hurwitz zeta-function. Denote by k
j jl j
the least common multiple of periods k ,...,k . Let B be the matrix consisting of
j1 jl(j) j
REMARKSON THEMIXEDJOINTUNIVERSALITY 7
coefficients b from theperiodicsequences B , l =1,...,l(j),m=1,...,k ,i.e.,
mjl jl j
b b ... b
1j1 1j2 1jl(j)
b b ... b 
B = 2j1 2j2 2jl(j) .
j
... ... ... ...
 
 
b b ... b 
 kjj1 kjj2 kjjl(j)
The functional independence for the above collection of periodic Hurwitz zeta-func-
tions was proved by A. Laurincˇikas in [7, Theorem 3] under a certain matrix condi-
tion. The proof is based on the joint universality theorem among periodic Hurwitz zeta-
functionsprovedbyJ. Steudingin[14].
Fortheproofof mixedjointfunctionalindependence, we may adoptthe methoddeve-
loped in series of works by A. Laurincˇikas and his colleagues (see, for example, [1], [8],
[13]). Then itispossibletoobtainthefollowinggeneralizationofTheorem 1.
Theorem4. Supposethata ,...,a arealgebraicallyindependentoverQ,rankB =l(j),
1 r j
1 ≤ j ≤ r, and j (s) belongs to the class S. Let the function F : CN(l +1) → C be a
j
continuousfunctionforeach j=0,...,n. Supposethatthefunction
e
n
(cid:229) sj·F j (s),j ′(s),...,j (N−1)(s),
j
(cid:18)
j=0
z (s,a ;B ),z ′(s,a ;B ),...,z (N−1)(s,a ;B ),...,
1 11 1 11 1 11
z (s,a ;B ),z ′(s,a ;B ),...,z (N−1)(s,a ;B ),...,
1 1l(1) 1 1l(1) 1 1l(1)
z (s,a ;B ),z ′(s,a ;B ),...,z (N−1)(s,a ;B ),...,
r r1 r r1 r r1
z (s,a ;B ),z ′(s,a ;B ),...,z (N−1)(s,a ;B )
r rl(r) r rl(r) r rl(r)
(cid:19)
is identicallyzero. Then F ≡0for j=0,...,n.
j
Thistheoremisaconsequenceofthefollowingmixedjointuniversalitytheorem,which
is a generalization of Lemma2. This theorem is also an analogue of a result of J. Genys,
R. Macaitiene˙, S. Racˇkauskiene˙ and D. Šiaucˇiu¯nas [1, Theorem 3], which treats the case
that j (s)isreplaced byz (s).
Theorem5. Supposethata ,...,a arealgebraicallyindependentoverQ,rankB =l(j),
1 r j
1≤ j ≤r, and j (s) belongs to the class S. Let K be a compact subset of D(s ∗,1), K
1 2jl
be compact subsets of D(1,1), all of which with connected complements. Suppose that
2 e
f ∈Hc(K )and f ∈Hc(K ). Then, foreverye >0,
1 0 1 2jl 2jl
1
liminf m t ∈[0,T]: sup|j (s+it )− f (s)|<e ,
1
T→¥ T (cid:26) s∈K1
max max sup |z (s+it ,a ;B )− f (s)|<e >0.
j jl 2jl
1≤j≤r1≤l≤l(j)s∈K2jl (cid:27)
8 ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
Remark1. Considerthecasewhenalll(j)=1,1≤ j≤r. WriteB =B . Inthiscase,
j1 j
the condition rankB =l(j) trivially holds. Therefore the joint universalityand the func-
j
tional independence among the functions j (s), z (s,a ,B ),...,z (s,a ,B ) are valid
1 1 r r
withoutany matrixcondition,onlyundertheassumptionthat a ,...,a are algebraically
1 r
independentoverQ. WhenA.Laurincˇikasstartedhisstudyontheuniversalityofperiodic
Hurwitz zeta-functions, he assumed various matrix-type conditions, but finally, the joint
universalityamongperiodicHurwitzzeta-functionswithoutanymatrixconditionwases-
tablished by A. Laurincˇikas and S. Skerstonaite˙ in [9, Theorem 3]. Our Theorems 4 and
5 includethe“mixed”generalizationofthistheorem.
5. PROOF OF THEOREMS 4 AND 5
In the proof of Theorems 4 and 5, the crucial role is played by a mixed joint limit
theorem in the sense of weakly convergent probability measures in the space of analytic
functions.
5.1. A generalized mixed joint limit theorem. Let D be an open subset of D(s ∗,1),
1
andD beanopensubsetofD(1,1). ForanysetS,byB(S)wedenotethesetofallBorel
2 2
subsets of S. For any region D, denote by H(D) the set of all holomorphic functions on
D. LetH betheCartesian productofl +1 suchspaces, i.e.
H =H(D )×H(D )×...×H(D ).
1 2 2
l
| {z }
Moreover,let
¥
W = (cid:213) g and W = (cid:213) g ,
1 p 2 m
p∈P m=0
whereg =g forall p∈P,g =g forall m∈N , andg ={s∈C:|s|=1}, anddefine
p m 0
W =W ×W ×...×W
1 21 2r
where W =W for all j =1,...,r. Then by the Tikhonovtheorem W is a compact topo-
2j 2
logicalAbeliangroupalso. Thenwehavetheprobabilityspace(W ,B(W ),m ). Herem
H H
is the product of Haar measures m ,m ,...,m , where m is the probability Haar
H1 H21 H2r H1
measure on (W ,B(W )) and m is the probability Haar measure on (W ,B(W )),
1 1 H2j 2j 2j
j=1,...,r. Letw (p) betheprojectionofw ∈W to g , and,forevery m∈N, define
1 1 1 p
w (m)= (cid:213) w (p)a
1 1
pa km
with respect to the factorization of m to primes. Denote by w (m) the projection of
2j
w ∈W to g ,m∈N , j=1,...,r.
2j 2j m 0
REMARKSON THEMIXEDJOINTUNIVERSALITY 9
For brevity, write a = (a ,...,a ), B = B ,...,B ,···,B ,···,B , s =
1 r 11 1l(1) r1 rl(r)
(s ,s ,...,s ,...,s ,...,s )∈Cl +(cid:0)1, and (cid:1)
1 211 21l(1) 2r1 2rl(r)
Z(s,a ;B)=(j (s ),z (s ,a ;B ),...,z (s ,a ;B ),
1 211 1 11 21l(1) 1 1l(1)
...,z (s ,a ;B ),...,z (s ,a ;B )).
2r1 r r1 2rl(r) r rl(r)
Let w =(w ,w ,...,w )∈W . Define theH-valued random element Z(s,a ,w ;B) on
1 21 2r
theprobabilityspace(W ,B(W ),m )by theformula
H
Z(s,a ,w ;B)=(j (s ,w ),z (s ,a ,w ;B ),...,z (s ,a ,w ;B ),
1 1 211 1 21 11 21l(1) 1 21 1l(1)
...,z (s ,a ,w ;B ),...,z (s ,a ,w ;B )),
2r1 r 2r r1 2rl(r) r 2r rl(r)
where
j (s,w )= (cid:229)¥ cmw 1(m), s∈D(s ∗,1),
1 ms
m=1
and
z (s,a ,w ;B )= (cid:229)¥ bmjlw 2j(m), s∈D 1,1 , j=1,...,r, l =1,...,l(r),
j 2j jl (m+a )s 2
m=0 j (cid:0) (cid:1)
respectively. Theseseriesareconvergentforalmostallw ∈W andw ∈W , j=1,...,r.
1 1 2j 2j
DenotebyP thedistributionoftherandomelementZ(s,a ,w ;B),i.e.,
Z
P (A)=m (w ∈W :Z(s,a ,w ;B)∈A), A∈B(H).
Z H
Nowwearereadytostateourmixedjointlimittheoremforthefunctionsj (s),z (s,a ;B ),
1 11
...,z (s,a ;B )as alemma.
r rl(r)
Lemma 6. Suppose that the numbers a ,...,a are algebraically independent over Q,
1 r
and j ∈S. Then theprobabilitymeasureP defined by
T
e 1
P (A)= meas{t ∈[0,T]:Z(s+it ,a ;B)∈A}, A∈B(H), (6)
T
T
converges weakly toP as T →¥ .
Z
When r =1 and l(1)=1, this lemma is Theorem 2.1 from [5]. On the other hand, the
same type of limit theorem with j replaced by the Riemann zeta-function is given in [1,
Theorem4]. Theproofoftheabovelemmaisquitesimilartothatofthoseresults,sohere
wegiveaverybriefoutline.
Let j (s), j (s,w ), z (s,a ;B), z (s,a ,w ;B) be the same as in [5, Section 3], and
n n 1 n n 2
define
b b
Z (s,a ;B)=(j (s ),z (s ,a ;B ),...,z (s ,a ;B ),
n n 1 n 211 1 11 n 21l(1) 1 1l(1)
...,z (s ,a ;B ),...,z (s ,a ;B ))
n 2r1 r r1 n 2rl(r) r rl(r)
10 ROMAKACˇINSKAITE˙ ANDKOHJIMATSUMOTO
and
Z (s,a ,w ;B)=(j (s ,w ),z (s ,a ,w ;B ),...,z (s ,a ,w ;B ),
n n 1 1 n 211 1 21 11 n 21l(1) 1 21 1l(1)
...,z (s ,a ,w ;B ),...,z (s ,a ,w ;B )),
b n b2r1 r 2r r1b n 2rl(r) r 2r rl(rb)
where w = (w ,w ,...,w ) ∈ W . bThe probability measuresbP , P and P on H
1 21 2r T,n T,n T
are defined similarly as P in (6), with replacing Z(s+it ,a ;B) by Z (s+it ,a ;B),
b b b bT b n b
Z (s+it ,a ,w ;B) and Z(s+it ,a ,w ;B), respectively. As an analogueof [5, Lemma3]
n
or [1, Lemma 2], we can show that both measures P and P converge weakly to a
T,n T,n
b b
certain probability measure P as T →¥ . (In the course of the proof, a key is Lemma 1
n b
from [1], which is based on the assumption that a ,...,a are algebraically independent
1 r
overQ.)
Next we need an approximation lemma in the mean value sense, similar to [5, Lem-
ma 3.3]. In the proof of [5, Lemma 3.3], we used the result of [3]. The desired approxi-
mation can be shown by using, instead of [3], mean value results given in [10, (2.3),
(2.5)].
Then we can prove that both P and P converge weakly to a certain probability mea-
T T
sureP. Thisisananalogueof[5,Lemma3.4]or[1,Lemma5]. Finallywecanshowthat
b
P=P , by theusual ergodicargument. ThiscompletestheproofofLemma6.
Z
5.2. Completion ofthe proof. Now wecompletetheproofof Theorems 4 and 5. Here-
after we assume that a ,...,a are algebraically independent over Q, rankB = l(j),
1 r j
1≤ j ≤r, and j (s)belongsto theclass S.
Let Sj betheset ofall f ∈H(D1)whiech isnon-vanishingon D1, orconstantly≡0 on
D .
1
l
Lemma 7. Thesupportof themeasurePZ isSj ×H(D2) .
This can be shown just analogously to [1, Theorem 5]. Then Theorem 5 follows from
Lemmas6 and 7by thestandard argument;seeagain[1].
Next,as a generalizationofLemma3, wecan showthat theimageof themapu:R→
C(l +1)N bytheformula
u(t)= j (s +it),j ′(s +it),...,j (N−1)(s +it),
(cid:18)
z (s +it,a ;B ),z ′(s +it,a ;B ),...,z (N−1)(s +it,a ;B ),...,
1 11 1 11 1 11
z (s +it,a ;B ),z ′(s +it,a ;B ),...,z (N−1)(s +it,a ;B ),...,
1 1l(1) 1 1l(1) 1 1l(1)
z (s +it,a ;B ),z ′(s +it,a ;B ),...,z (N−1)(s +it,a ;B ),...,
r r1 r r1 r r1
z (s +it,a ;B ),z ′(s +it,a ;B ),...,z (N−1)(s +it,a ;B )
r rl(r) r rl(r) r rl(r)
(cid:19)