Distin t zeros of L-fun tions
by
E. Bombieri (Prin eton, N.J.)and A. Perelli (Genova)
1. Introdu tion. Let L
1
(s) and L
2
(s) be two \independent" L-fun -
tions, where themeaning of \independent" willbe lari edlater on. Sin e
theL-fun tionsaredeterminedbytheirzeros,wemayexpe tthatL
1
(s)and
L
2
(s)have few ommon zeros. Thisproblemappearsto bevery diÆ ultat
present,thereforewemayasktheeasierquestionofgettingafairquantityof
distin tzeros of su h fun tions. Inthispaperwe showthat, under suitable
onditions,L
1
(s) and L
2
(s) have a positiveproportionofdistin tzeros.
We state our results in the moderately general setting of Bombieri{
Hejhal's paper [1℄, whi h also provides thebasi ingredients of the present
paper. Moreover, we willwork outour maintool,Theorem 2below,in the
ase of several L-fun tions. Hen e, for a given integer N 2, we onsider
N fun tions L
1
(s);:::;L
N
(s) satisfyingthefollowingbasi hypothesis.
HypothesisB.(I)Ea hfun tionL
j
(s)hasanEulerprodu toftheform
L
j (s)=
Y
p d
Y
i=1 (1
ip p
s
) 1
withj
ip jp
for some xed 0<1=2 andi=1;:::;d.
(II) For every ">0 we have
X
px d
X
i=1 j
ip j
2
x 1+"
:
(III) The fun tions L
j
(s) have an analyti ontinuation to C as mero-
morphi fun tions of nite order with a nite number of poles, all on the
line =1,and satisfy a fun tional equationof the form
(s)="(1 s);
where(s)=Q s
Q
m
i=1 (
i s+
i
), Q>0,
i
>0,Re
i
0 andj"j=1.
1991 Mathemati sSubje tClassi ation: Primary11M41.
(IV ) The oeÆ ients a
j
(p) of the Diri hlet series
L
j (s)=
1
X
n=1 a
j (n)n
s
satisfy
X
px a
j (p)a
k (p)
p
=Æ
jk n
j
loglogx+
jk +O
1
logx
for ertain onstants n
j
>0.
Weexpli itlyremarkthatallthedatainvolvedinHypothesisB on ern-
inga fun tionL
j
(s) maydependon j. We also remarkthatthe onditions
of Hypothesis B may be somewhat relaxed (see Selberg [10℄) in order to
dedu e ourresultsbelow.
Wereferto Se tion3of[1℄ forathoroughdis ussionofHypothesisB,of
itsstandard onsequen esand ofseveralexamplesoffun tionssatisfyingit.
HerewepointoutonlythatB(II) impliesthatboththeDiri hletseriesand
theEulerprodu tof L
j
(s) onverge absolutelyfor >1,B(I) ensuresthat
L
j
(s) 6=0 for >1 and B(III) gives riseto thefamiliar notions of riti al
strip, riti allineand trivialand non-trivialzeros. Moreover, writing
j
= m
X
i=1
i
;
N
j
(t)=jf%:L
j
(%)=0; 0Re%1 and 0Im%tgj
and
S
j (t)=
1
argL
j
(1=2+it);
forsuÆ ientlylargetwe have
(1) N
j (t)=
j
tlogt+
j t+
0
j +S
j
(t)+O(1=t)
with ertain onstants
j and
0
j .
Condition B(IV), introdu ed bySelberg [10℄, plays a spe ialrole, sin e
it provides a form of \near-orthogonality" of the fun tions L
j
(s); the \in-
dependen e" alluded to at the beginning of the se tion omes from this
\near-orthogonality". For instan e, B(IV) implies that L
1
(s);:::;L
N (s)
arelinearlyindependentoverC; seeBombieri{Hejhal[1℄and Ka zorowski{
Perelli[7℄forfurther resultsinthisdire tion.
We expe t that the fun tions L
j
(s) satisfy the Generalized Riemann
Hypothesis. Asa substitute of itinour arguments, we willinstead assume
thefollowingdensityestimate. Let
N (;T)=jf%:L (%)=0; Re% and jIm%jTgj:
Hypothesis D.Thereexists 0<a<1 su h that
N
j
(;T) T
1 a( 1=2)
logT
uniformly for 1=2 and j=1;:::;N.
The mainpoint in introdu ingHypothesis D is that, unlike the Gener-
alizedRiemann Hypothesis,it an beveri edin manyinteresting ases. In
fa t, it has been proved by Selberg [9℄ for the Riemann zeta fun tion, by
Fujii[5℄ for Diri hlet L-fun tions, and byLuo[8℄ in themore diÆ ult ase
of L-fun tionsatta hed to ertainmodularforms.
In order to state our main result, we de ne the ounting fun tion
D(T;L
1
;L
2
) of the distin t non-trivial zeros, ounted with multipli ity,of
two fun tions L
1
(s) and L
2 (s) as
D(T;L
1
;L
2 )=
X
0Re%1
0Im%T
max(m
1
(%) m
2 (%);0);
where%runsoverthezerosofL
1 (s)L
2
(s)andis ountedwithoutmultipli -
ity. We also de ne
D(T)=D(T;L
1
;L
2
)+D(T;L
2
;L
1 )=
X
0Re%1
0Im%T jm
1
(%) m
2 (%)j;
withthe same onvention about%.
Our mainresult is
Theorem 1.Let L
1
(s) and L
2
(s) satisfy Hypotheses Band D and sup-
pose that
1
=
2
. Then
D(T;L
1
;L
2
) TlogT:
Clearly,thesame lower boundholdsforD(T;L
2
;L
1
) andD(T)too.
The rst resultof thistypehasbeenobtainedbyFujii[6℄in the aseof
twoprimitiveDiri hletL-fun tions,bymeansofSelberg'smomentsmethod.
The problem of ounting strongly distin t zeros, i.e., zeros pla ed at dif-
ferent points, appears to be more diÆ ult, and the best result is due to
Conrey{Ghosh{Gonek [3℄, [4℄. They deal withthis problem, inthe ase of
two primitive Diri hlet L-fun tions, by onsidering the more diÆ ultques-
tion of getting simple zeros of L(s;
1 )L(s;
2
), and show that there are
T 6=11
su h zeros up to T. Moreover, if the Riemann Hypothesis is as-
sumedforone ofthetwofun tions,thenapositiveproportionofsu hzeros
is obtained. However, the te hniques in[3℄ and [4℄ do not extend to over
the ase of moregeneral L-fun tions, su hasGL
2
L-fun tions.
Let us all oprime two fun tions in Selberg's lass S (see [10℄) ea h
su h that there are no ommon fa tors of su h fa torizations. Assuming
Selberg's Conje tures 1.1 and 1.2 in [10℄, we see that B(IV) holds for o-
prime fun tions. Hen e, assuming Hypothesis D for every fun tion in S,
we may regardthelowerboundinTheorem 1,inthe ase of oprimefun -
tions, as a onsequen e of Selberg's onje tures. Another onsequen e of
Selberg's onje turesisthat S hasuniquefa torization (see Conrey{Ghosh
[2℄). Weremarkherethat thelatter onsequen eof Selberg's onje turesis
easilyimplied by a very weak form of theformer. Pre isely,assumingthat
two oprimefun tions inS have D(T) 1 forsuÆ iently large T, we get
theunique fa torization in S. In fa t, the assumption impliesthat two o-
primefun tions arene essarily distin t,andthis learlyimpliesthe unique
fa torization.
Theorem 1appears to bethe limitofour method,althoughmu h more
is expe ted to hold. Forinstan e, if L
1
(s) and L
2
(s) are distin t primitive
fun tions, we expe t that almost all zeros of L
1
(s) and L
2
(s) are distin t,
i.e.,
D(T)
1 +
2
TlogT;
inwhi h ase almostall zeros area tually stronglydistin t,oreven that
D(T)=N
1
(T)+N
2
(T)+O(1);
i.e., L
1
(s) and L
2
(s) have O(1) ommon non-trivialzeros.
The proof of Theorem 1 is based on Bombieri{Hejhal's [1℄ variant of
Selberg's[9℄moments method,whi hleadsinamore dire twayto thedis-
tributionfun tionforthelogL
j
(1=2+it)(see TheoremBof[1℄). Although
we ould followa variant more inthe spiritof Selberg [9℄ and Fujii[6℄, we
will prove Theorem 1 by means of a short intervals analog of the above
mentionedTheoremB,whi hwe believeto beof interestinitself.
Let M 10, writeh=M=logT and
V
j (t)=
logL
j
(1=2+i(t+h)) logL
j
(1=2+it)
(2n
j
logM) 1=2
;
and let
T
denote theasso iatedprobabilitymeasureon C N
,de ned by
(2)
T ()=
1
T
jft2[T;2T℄:(V
1
(t);:::;V
N
(t))2gj
for every open set C N
. Moreover, let e
kzk 2
denote the gaussian
measure on C N
and let d!be theeu lideandensityon C N
.
Theorem 2. Let L
1
(s);:::;L
N
(s) satisfy Hypotheses B and D and let
M =M(T)!1withM (logT)=loglogT asT !1. Then,asT !1,
tends to the gaussian measure withasso iated density e
kzk 2
d!.
We remark thatwe an easily get a slight variant of Theorem 2,where
h=M=logt and M =M(t)! 1with M log 1 "
t ast!1. Therefore,
ifwe separatetheV
j
(t)into theirrealand imaginaryparts,Theorem2 an
beexpressed bysayingthat thefun tions
log
L
j 1
2
+i t+ M
logt
log
L
j 1
2 +it
(2n
j
logM) 1=2
; j=1;:::;N;
and
argL
j 1
2
+i t+ M
logt
argL
j 1
2 +it
(2n
j
logM) 1=2
; j=1;:::;N;
be ome distributed, in the limit of large t, like independent random vari-
ables, ea h having gaussian density exp( u 2
)du, provided M ! 1 with
M log 1 "
tas t!1.
A knowledgments. The se ond named author wishes to thank the
InstituteforAdvan edStudy forits hospitalityand forprovidingex ellent
working onditions.
2. Basi lemmas. Inthisse tionwefollowtheargumentsinSe tion5
of Bombieri{Hejhal[1℄. For>1and j=1;:::;N we write
logL
j (s)=
1
X
n=1
j (n)
1 (n)n
s
;
1 (n)=
0; n=1;
(n)=logn; n2;
and denote by u(x) a real positive C 1
fun tion with ompa t support in
[1;e℄ andbyu(s)e its Mellin transform. Wealso write
v(x)= 1
\
x
u(t)dt
and assumethat u isnormalized sothatv(0)=1. We refer to Lemma1 of
[1℄ and theremarkfollowingitforrelevant propertiesof eu(s).
By (5.4) of [1℄ we havethe approximateformula
logL
j
(1=2+it)= 1
X
n=1
j (n)
1 (n)
n 1=2+it
v(e
(logn)=logX
) (3)
+ X
% 1
\
1=2 1
% s e
u(1+(% s)logX)d+O(1);
where jtjis suÆ ientlylarge and nottheordinate of azero of L
j
(s),where
2 X t 2
and where % runs over zeros of L
j
(s) with 0 Re% 1. We
write(3)as
logL
j
(1=2+it)=D
j
(1=2+it;X)+R
j
(1=2+it;X)
whereD (1=2+it;X) is theDiri hletserieson theright handsideof (3).
FromLemma 3 of[1℄ we immediatelygetour rstbasi lemma.
Lemma 1. Assume Hypotheses B and D, and let 2 X T a=2
and T
suÆ iently large. Then for j=1;:::;N we have
2T
\
T jR
j
(1=2+it;X)jdt T logT
logX :
Our se ond basi lemma is a short intervals analog of Lemma 6 of [1℄,
i.e., themixedmomentsofthedi eren es oftheD
j
(1=2+it;X). Sin e the
proofof Lemma2belowfollowsthatof Lemma6of [1℄,we willonlysket h
it. ForsuÆ ientlylargeM,writeh=M=logT and
j
(t)=D
j
(1=2+i(t+h);X) D
j
(1=2+it;X):
Moreover, let k
j
0 and l
j
0, j = 1;:::;N, be integers and let us
abbreviatek=(k
1
;:::;k
N ),K
j
=k
1
+:::+k
j
,K =K
N
and similarlyfor
l;L
j
and L. We also writek!= Q
N
j=1 k
j
!.
We state here the basi estimate we will repeatedly use in the proof of
Lemma2. ForX3 we have
(4) X
p a
j (p)a
k (p)
p
v(e
(logp) =logX
) 2
je
ihlogp
1j 2
=Æ
jk 2n
j log
+
h
2 logX
+O(1)
uniformly for h 1=loglogX, where log +
x = max (logx;0). In fa t,
je
ihlogp
1j = 4sin 2
((h=2)logp) and hen e (4) follows from B(IV) by
partialsummation(see also (3.8) of [1℄).
Lemma 2. Assume Hypothesis B and let X T
1=(K+L+1 )
and M
(logT)=loglogX. Write
j (t)=
1
X
n=1 b
j (n)
n 1=2+it
; b
j
(n)=
j (n)
1 (n)v(e
(logn) =logX
)(e
ihlogn
1):
Then
2T
\
T N
Y
j=1 (
j (t))
k
j
(
j (t))
l
j
dt =Æ
k;l k!T
N
Y
j=1
2n
j log
+
M
2 logT
logX
k
j
+O
T
log +
M
2 logT
logX
(K+L 1)=2
:
Proof. Wemay learlyassumethatK+L1. Fornotationalsimpli -
ity,weabbreviate
j
=
j
(t). Sin e
j
issupported atprime powers only,
we splitit as
j
= 0
+ 00
where 0
j
rangesover primesp and 00
j
over primepowersp r
,r2. Then,
a ordingly,we get
(5)
N
Y
j=1 (
j )
k
j
(
j )
l
j
= N
Y
j=1 (
0
j )
k
j
(
0
j )
l
j
+R (t);
where, asintheproof of Lemma6 of[1℄,
(6)
2T
\
T
jR (t)jdt 2T
\
T j
00
j1 jj
0
j2 j
K+L 1
dt+ 2T
\
T j
00
j3 j
K+L
dt
fora suitable hoi e of j
1
;j
2 and j
3 .
Sin e e
ihlogn
1 1, by(5.14) of [1℄ we have
(7)
2T
\
T j
00
j j
2(K+L)
dt T
forj=1;:::;N,providedX T
1=(K+L+1 )
.
By Montgomery{Vaughan'smean-valuetheoremforDiri hletpolynomi-
als (see,e.g.,Lemma 4 of[1℄) wehave
2T
\
T j
0
j j
2(K+L)
dt=T X
jB 0
(n)j 2
n
+O
X
jB 0
(n)j 2
;
where
B 0
(n)=
X
p1:::p
K+L
=n b
j (p
1 ):::b
j (p
K+L ):
Sin e
j
(p)=a
j
(p)and
1
(p)=1,from (5.16) of[1℄ and (4)we get
X
jB 0
(n)j 2
n
(K+L)!
X
p jb
j (p)j
2
p
K+L
log +
M
2 logT
logX
K+L
:
Moreover, from (5.17) of [1℄ we obtain
X
jB 0
(n)j 2
X
(1+")(K+L)
;
and hen e
(8)
2T
\
T j
0
j j
2(K+L)
dt T
log +
M
2 logT
logX
K+L
providedX T
1=(K+L+1 )
.
From (6){(8)and Holder'sinequalitywe get
(9)
2T
\
T
jR (t)jdt T
log +
M
2 logT
logX
(K+L 1)=2
1=(K+L+1 )
In order to treat themainprodu t on theright handsideof (5)we use
again Lemma4 of[1℄. We abbreviaten=(n
1
;:::;n
K ),
b(n;k)= N
Y
j=1 K
j
Y
r=K
j 1 +1
b
j (n
r
) and B(n;k)= X
n
1 :::n
K
=n
b(n;k);
and asin(5.18) of [1℄ we have
2T
\
T N
Y
j=1 (
0
j )
k
j
(
0
j )
l
j
;dt =T X
B(n;k)B(n;l)
n (10)
+O
X
jB(n;k)j 2
1=2
X
jB(n;l)j 2
1=2
;
where the sums are restri ted to n of type n = p
1 :::p
K
for k and n =
q
1 :::q
L
for l; here p and q denote prime numbers. By a variant of the
argument leading to (8)we seethat
(11)
X
jB(n;k)j 2
1=2
X
jB(n;l)j 2
1=2
T
log +
M
2 logT
logX
(K+L 1 )=2
providedX T
1=(K+L+1 )
.
In view of (5), (9), (10) and (11), to omplete theproof of Lemma 2 it
suÆ es to showthat
X
B(n;k)B(n;l)
n
=Æ
k;l k!
N
Y
j=1
2n
j log
+
M
2 logT
logX
k
j
(12)
+O
log +
M
2 logT
logX
(K+L 1 )=2
:
IfK 6=Lthere isnothingtoprove,sin eB(n;k)B(n;l)=0forevery n; we
an therefore assume K =L1 and pro eed by indu tionasinLemma 6
of [1℄.
If K =1, (12)follows immediately from (4). Supposenow that K 2.
Arguing again as inLemma 6 of [1℄ and using (3.8) of [1℄, we see that the
ontribution to the left hand side of (12) oming from n's whi h are not
square-free is
log +
M
2 logT
logX
(K+L 1 )=2
:
In order to deal with the remaining part of the sum on the left hand side
take into a ount thefa tor e
ihlogn
1 in ourde nition of theb
j
(n). In
thiswaywesee that(12) holdsforanyK 1,and Lemma 2is proved.
We remarkthat we an easily obtain a version of Lemma 2 with h re-
pla edbyM=logt, providedanadditional errorterm
O
T(loglogX) K+L
MlogX
log 2
T
isaddedinthestatementofLemma2. Weleaveitsveri ationtothereader.
3. Proof of theorems. Theproofof Theorem2 follows loselythatof
TheoremBof [1℄. LetM !1 asT !1 and hoose
logX=
logT
(logM) 1=4
;
sothat
log +
M
2 logT
logX
logM;
logX
logT
=(logM) 1=4
; X=T o(1)
:
Moreover, let
U
j
(t)=(2n
j
logM) 1=2
j (t)
and e
T
be theasso iatedprobabilitymeasure on C N
,de ned asin(2).
Then, assuming that M (logT)=loglogT and arguing exa tly as in
theproofof TheoremBof [1℄,from Lemma2 we see thate
T
onverges, as
T ! 1, to the gaussian measure e
kzk 2
. Also, from Lemma 1 we easily
dedu e that
1
T 2T
\
T jV
j
(t) U
j
(t)jdt (logM) 1=4
;
andhen e
T
onvergestothesamegaussianmeasure, ompletingtheproof.
The proofofTheorem1 isby ontradi tion. LetT
bea sequen ealong
whi h
D
:=D(2T
;L
1
;L
2
) D(T
;L
1
;L
2
)=o(T
logT
):
We set
(13) M
=min
logT
loglogT
; r
T
logT
1+D
:
Then M
! 1 and M
(logT
)=loglogT
, sothat Theorem2 isappli-
able to L
1 ,L
2
and thesequen e T
,M
.
Write
h
=M
=logT
;
N (t;h
)=(N
1 (t+h
) N
1
(t)) (N
2 (t+h
) N
2 (t));
(t;h )=(S (t+h ) S (t)) (S (t+h ) S (t))
and observe that (1)and
1
=
2 imply
(14)
N (t;h
)=
S (t;h
)+O
M
logT
uniformlyfort2[T
;2T
℄.
Forj=1;2and t2[T
;2T
℄we have
(15) ImV
j (t)=
(2n
j logM
)
1=2 (S
j (t+h
) S
j (t)):
Thusfrom (14) and (15)we see thatift2[T
;2T
℄is su h that
(16) ImV
2
(t)<0 and ImV
1
(t)>1;
then
N (t;h
)=
1
(2n
1 logM
)
1=2
ImV
1 (t)
1
(2n
2 logM
)
1=2
ImV
2
(t)+O(M
=logT
)
1
(2n
1 logM
)
1=2
+O(M
=logT
):
Denote byE
thesetof t2[T
;2T
℄forwhi h(16) holds.
In order to geta lower boundforjE
j,we onsidertheset
=f(z
1
;z
2 )2C
2
:Imz
1
>1 and Imz
2
<0g;
sothat
(17) jE
j=T
T
():
From Theorem2 we obtain
(18) lim
!1
T
()=
\
e
kzk 2
d! 1:
From (15), (17) and (18) we see that jE
j T
, and hen e we dedu e
theexisten e of T
=h
values t
r 2[T
;2T
℄, withjt
r t
s jh
if r6=s,
su h that
N (t
r
;h
)
1
(2n
1 logM
)
1=2
+O(M
=logT
):
Therefore
(19) D
X
r
N (t
r
;h
)
p
logM
M
T
logT
:
Nowre all that
(13) M
=min
logT
loglogT
; r
T
logT
1+D
:
If in(13)we have M
=(logT
)=loglogT
wemust also have
D
T
(loglogT
)
2
logT
;
while (19) gives D
T
(loglogT
)
3=2
, a ontradi tion. The other al-
ternative in (13) gives M
= p
(T
logT
)=(1+D
) , whi h substituted in
(19) shows that D
T
logT
; this ontradi ts our assumption D
=
o(T
logT
).
The proof of Theorem1is omplete.
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S hoolofMathemati s DipartimentodiMatemati a
InstituteforAdvan edStudy ViaDode aneso35
Prin eton,NewJersey08540 16146 Genova
U.S.A. Italy
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