ON THE SELBERG CLASS OF DIRICHLET
SERIES SMALL DEGREES
J B Conrey
A Ghosh
Introduction
In the study of Dirichlet series with arithmetic signi cance there has app eared
through the study of known examples certain exp ectations namely i if a func
tional equation and Euler pro duct exists then it is likely that a type of Riemann
hypothesis will hold ii that if in addition the function has a simple p ole at the
p oint s then it must b e a pro duct of the Riemann zeta function and another
Dirichlet series with similar prop erties and iii that a type of converse theorem
holds namely that all such Dirichlet series can b e obtained by considering Mellin
transforms of automorphic forms asso ciated with arithmetic groups Guided by
these ideas consider the class S of Dirichlet series introduced by Selb erg a
Dirichlet series
X
a
n
F s
s
n
n
is in S provided that it satis es the following hypotheses
m
Analyticity s F s is an entire function of nite order for some
non negative integer m
Ramanujan Hyp othesis a n for any xed
n
Functional equation there must b e a function s of the form
F
k
Y
s
w s s Q
i i F
i
where j j Q w and such that
i i
s s F s
F
satis es
s s
where s s
Research of the second author supp orted in part by the Alfred P Sloan Foundation Research
of b oth authors supp orted in part by a grant from the NSF
Typeset by A S T X
M E
J B CONREY A GHOSH
Euler pro duct a and
X
b
n
log F s
s
n
n
where b unless n is a p ositive p ower of a prime and b n for some
n n
One is interested in classifying functions in this class and determining if prop
erties i iii hold With this in mind it is clear that one is primarily interested
in the primitive elements in this class namely the ones that cannot b e written as a
pro duct of two or more non trivial members of S detailed de nitions can b e found
in Section For the problem of classi cation one needs to de ne the degree d
F
of F S as
k
X
d w
F i
i
allowing for the p ossibility d if the pro duct of gamma functions is empty
F
Then S d will denote the subset of S consisting of all functions of degree d
Remarks
s
We do not consider generalised Dirichlet series here as the example
would b e a candidate that would violate the Riemann hypothesis
The condition that there b e at most one p ole and that at s is natural
since if we exp ect condition ii to hold and if F has a nite number of p oles
then for each p ole F s would have the Riemann zeta function suitably
shifted as a factor So the p oles would lie on the line s otherwise
F would have zeros corresp onding to the shifted zeros of the Riemann
Thus we may write zeta function not on the line s
R
Y
s it F s E s
j
j
for some real numbers t and an entire function E We then nd nothing
j
new by allowing these p oles into our condition and instead fo cus on
functions with at most one p ole normalised to b e at s
In the functional equation the restriction may b e explained in
i
the following way Supp ose there exists an arithmetic subgroup of SL R
together with a Maass cusp form that corresp onds to an exceptional eigen
value and also assume that the Ramanujan Petterson conjecture holds
Then the L function that is asso ciated with the Maass form has a func
tional equation with a satisfying but that violates the Riemann
i i
hypothesis This suggests that a restriction of the type is appro
i
priate
Condition corresp onds to the familiar notion of Euler pro duct In fact
it is easy to verify that if F S then the co e cients a of its Dirichlet
n
series are multiplicative i e a a a if m and n are relatively prime
mn m n
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
Consequently F has an Euler pro duct expansion
Y
F s F s
p
p
where the pro duct is over primes and
X
a k
p
F s
p
k s
p
k
From the viewp oint of automorphic L functions it is natural to put addi
tional restrictions on F s namely that for almost all primes F s is a
p p
s
p olynomial in p of degree indep endent of p However in this pap er we
will not need such a restriction
The condition turns out to b e surprisingly imp ortant Note that
s
would violate the Riemann hypothesis as the example s
shows It also plays a crucial part in Theorem
In Selb erg several conjectures were made concerning functions in S These
are
Regularity of distribution There is an integer n asso ciated to each F such
F
that
X
ja j
p
n log log x O
F
p
p x
Orthonormality If F and F are distinct and primitive then n and
F
X
a a
p
p
O
p
p x
GL twists If is a primitive Dirichlet character and if F and F S
then
k
Y
F F
i
i
where the F are primitive implies that
i
k
Y
F F
i
i
are also primitive elements of S and the F
i
Selb erg also conjectures that the Riemann hypothesis holds for this class of
functions i e if F S then all non trivial zeros of F s see Sec are on the
line However we will not make use of this conjecture anywhere in this
pap er In particular in section where we prove some consequences of Selb erg s
conjectures we do not use this one in our pro ofs
J B CONREY A GHOSH
Other fundamental questions also arise For example all instances of functions
in S have degrees that are integers and so one would like to know if S d is empty
when d is not an integer Similar questions arise as to the admissible values of Q
see Sec
Examples
Clearly s S as is L s i for a primitive Dirichlet character
Condition prevents s i from b eing in the class if
The Dedekind zeta function of a number eld of degree d is in S d as are
the ab elian L functions of primitive characters asso ciated with such Also
an Artin L function for an irreducible representation of the Galois group of
the number eld is in S provided that it is holomorphic i e that Artin s
conjecture holds
An L function asso ciated with a holomorphic newform on a congruence
subgroup of SL Z once it is suitably normalized is in S An L
function asso ciated with a non holomorphic newform is presumably also in
this class but condition has not b een established for such Also if such
a newform corresp onds to an exceptional eigenvalue then condition do es
not hold since
The Rankin Selb erg convolution of any two holomorphic newforms is in S
The symmetric square L function asso ciated with a holomorphic newform
on the full mo dular group is in S
One can show that a large class of functions b elong to S if one is willing to
accept other well known conjectures For instance Langland s conjectures
allow L functions asso ciated with symmetric p ower representations to b e
in S Indeed if L s is asso ciated with a cusp form on the mo dular group
say and
Y
p p
L s
s s
p p
p
b elongs in S and if L s denotes the m th symmetric p ower L function
m
m
Y Y
j m j s
p L s
m
p p
p
j
then the Langlands conjectures imply that L s b elongs to S for each
m
m and they are entire It was shown by Serre that this together with
the non vanishing on the line implies the Sato Tate conjecture on
the distribution of the arguments of the s It can then b e shown quite
p
easily that the Sato Tate conjecture implies that n for all m so that
L
m
Selb erg s conjectures would imply that all these L functions are primitive
It can b e shown that the existence of a continuous distribution function
for the arguments of the s leads to the statement the Sato Tate conjec
p
ture is true if and only if n for all m If one assumes only that the
L
m
n are integers then one can show that there are at most two p ossible
L
m
distributions the Sato Tate b eing one There are similar statements for
the Hasse Weil L functions assuming the Taniyama Weil conjectures and
Artin L functions assuming Artin s holomorphy conjecture
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
In this pap er we discuss in some detail the structure of S d for d some
consequences of Selb erg s conjectures and the structure of a sub class of S and
S
De nitions We reserve F and G to denote members of S
For F S the Q w and are not well de ned For example we could use
i i
the duplication formula
w s
w s w s w s
to obtain a di erent Q w and Also could b e replaced by without a ecting
i i
the hypotheses for F to b e in S However we shall show momentarily
Theorem If and are b oth admissible gamma factors for F then
s C s for some real constant C
Thus is well de ned up to a constant As this constant presents no particular
F
problem for the moment we will use the notation for one of the class of gamma
F
factors Later we will single out a particular choice Now we de ne some notions
Trivial zeros The p oles of s are at the numbers s n w i
F i i
k n which are all in From the functional equation it follows
that F has a zero at s of order
X
m
s
i n
n w
i i
where m is the order of the p ole of F at s i e m m and m if s
s s
These zeros are the trivial zeros of F and all other zeros are the non trivial zeros
We do not exclude the p ossibility that F has a trivial zero and a non trivial zero
at the same p oint
Degree We de ne the degree d of a gamma factor for F by the formula
k
X
w d
i
i
We allow for the p ossibility that d as would happ en if the pro duct of gamma
functions were empty We will see later that the function
F s
is the only degree function in S
Note that if s and s are two admissible gamma factors for F with
degrees d and d then d d For if we form the quotient of the two
functional equations for F say h we obtain
h s h s
Now the left side is regular and non zero in and the right side is regular and
nonzero in Hence h s is an entire non vanishing function If the degrees
J B CONREY A GHOSH
were di erent then h would have zeros or p oles Thus we may sp eak of the degree
d of F
F
Arguing further we give the pro of of Theorem We see that h is of order
by Stirling s formula Hence it has the shap e
as b
e
Then
a s b as b
e e
so that a a i e a i where is real Letting s it we see again by Stirling s
formula that
h it
h it
as t Therefore Thus Theorem follows
Primitive Next we say that F S is primitive if F F F with F F S
implies that F or F
Twists For a primitive Dirichlet character and an F S we de ne
X
a n
n
F s
s
n
n
We end this section with some useful lemmas
Lemma If F s S then F s has no zeros in and F s has no
p
zeros in
For
X
b k
p
log F s
p
k s
p
k
converges absolutely for by condition and so has no zeros in
Since the Euler pro duct converges absolutely for there can b e no zeros of F
in this half plane
Lemma Supp ose that F S and F s has a p ole or zero at s i where
is real Then the sums
X
a
p
i
p
p x
are unbounded as x
Proof We have
m
F s C s i
as s i i for some integer m Then
log F s m log
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
But by the b ound on the b in condition
n
X X X X
jb k j
b a
n p p
log F s O
s s k
n p p
p p
n
k
X
a
p
O
s
p
p
where the sums over p are for primes p Thus
X
a
p
m log
s
p
p
as Let
X
a
p
S x
i
p
p x
Assume that S x is b ounded Then
Z
X
a
p
x dS x
s
p
p
Z
S x x dx O
which is a contradiction
Non existence of functions with small degrees In this section we prove
Theorem If F S then F or d
F
R
to mean the integral Proof Supp ose that d Then using the notation
F
a b
from c i to c i for any c with a c b we have
X
nx
a e h x
n
n
Z
s
x s F s ds
i
P log x
K x
x
where P is a p olynomial and
n n
X
F n x
K x
n
n
n n
X
n F n x
F
n n
F
n
J B CONREY A GHOSH
is an entire function of x since
n
F
d n n
n A
n n
F
for some A
Thus we see that h x is analytic in the plane slit along the negative real axis
But h is p erio dic with p erio d i so in fact h has no singularities on the real axis
P
Thus h is entire Let H z a e nz h iz Now if y then
n
n
Z
ny
a e H x iy e nx dx
n
Both sides of this equation are entire functions of the complex variable y We
di erentiate twice with resp ect to y and set y
Z Z
jH x j dx H x e nx dx n a
n
Hence a n But then the Dirichlet series for F s is absolutely convergent
n
for In particular F is uniformly b ounded in But this is easily
seen to b e a contradiction if d as
s s
F
F s
s s
F F
for by Stirling s formula
s
F
d
c t
s
F
for some c as t
In the case d we argue slightly di erently Since H is entire we see that
its Fourier series expansion is a p ower series in e z which is entire so that it
is convergent in the whole plane This conclusion necessitates that the a must
n
b e small In fact the a will b e so small that the Dirichlet series for F will b e
n
absolutely convergent in the whole complex plane
Then the functional equation for F can b e viewed as an identity b etween abso
lutely convergent Dirichlet series We write the functional equation as
s
X X
a Q
n
s
a Q n
n
n n
n n
It follows that if a for some n then Q n is an integer Therefore Q is
n
an integer and a implies that n j Q Thus the Dirichlet series is really a
n
Dirichlet p olynomial Now if Q then F We assume that q Q
Note that a implies that
ja j Q q
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
Since a is a multiplicative function it follows that there is a prime p and a p ositive
n
r
integer r such that p jj q and
r
r
j p ja
p
Now the Euler factor corresp onding to the prime p is
r
X
j
a
p
F s
p
j s
p
j
and its logarithm is
X
j
b
p
log F s
p
j s
p
j
s
j j
Replace p by x a by A and b by B Then the Euler factor is
j j
p p
r
X
j
P x A x
j
j
and its logarithm is
X
j
B x log P x
j
j
Since A we can factor P as
r
Y
P x R x
i
i
Taking the logarithm of b oth sides here we obtain a formula for B
j
r
j
X
R
i
B
j
j
i
Now
r
Y
jR j Q
i
i
Therefore
max jR j p
i
i r
Then
1
j
r
j
X
R
j j
i
j
max jR j p jb j jB j
i j
p
i r
j
i
This contradicts the existence of a for which
b n n
J B CONREY A GHOSH
and so completes the pro of of the theorem
This metho d of pro of can b e used to show that if F S and d then
Q The mo di cation is that if d and Q then K is no longer
entire but is regular in a circle ab out the origin of radius Q where Q
is a p ositive function of Q which tends to as Q Then by p erio dicity
we see that the p ower series h exp x is regular apart from the negative real
axis in a strip Q where Q is a p ositive function of Q which tends to
as Q Then the series for h is convergent in this region which once again
forces the a to b e to o small to b e co e cients of a function in S
n
We remark that Bo chner has a theorem which is relevant here See also
Vigneras His result in our context is
Theorem Bo chner Fix Q w C for i k with
i i
P
k
w The number of linearly indep endent Dirichlet series F s which
i
i
satisfy a functional equation with a satisfying these requirements is
F
k
Y
w
i
w q
i
i
where q Q
His pro of involves Fuch s theorem on di erential equations and Polya s gap the
orem on singularities of p ower series
We remark that as a consequence of Theorem we have
Corollary Any function in S can b e factored into a pro duct of primitives
Proof This assertion follows from the additivity of the degree function d
FG
d d and the lower b ound that F d
F G F
As a second consequence we have
Corollary If F S and d then F is primitive
F
Consequences of Selb erg s conjectures In this section we assume Sel
b erg s conjectures and give some of the immediate consequences many of which
are mentioned in Selb erg s pap er
Prop osition If
e e e
1 2
k
F F F F
k
where the F are primitive then
i
n e e e
F
k
Proof It is clear that
k
X
a F e a F
p i p i
i
Then
k
X X
a F ja F j e ja F j a F
p i p p i p j
i
i
i j
whence the result follows by orthogonality
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
Prop osition If n then F is primitive
F
Proof This follows immediately from the rst Prop osition since if F had a factor
ization as ab ove then
k
X
n e
F
i
i
so that k and e
m
Prop osition If F s has a p ole of order m at s then s divides F
Proof Clearly it su ces to prove this assertion in the case that m and F
is primitive By Prop osition is primitive since n Take F F and
F in the orthogonality conjecture If F then that conjecture implies that
P
a F
p
O But this contradicts Lemma
p x
p
Prop osition S has unique factorization
Proof It su ces to show that if F is primitive and F j GG then F j G or F j G
Assume neither holds Supp ose that FF GG We express b oth sides of this
equation in terms of primitive functions as
g g f f
f
1 1
l k
G G F F F
l k
r
where F F and G are distinct primitive functions Multiply b oth sides by F
i i
and compute n for b oth sides
r f O r O
We have a contradiction as r
Prop osition Dedekind s conjecture If K is a nite extension of Q and
is the Dedekind zeta function of K then
K
L s s s
K
is entire
For S and has a simple p ole at s Hence it will b e divisible by in S
K
We remark that R Murty in work to app ear has shown that Artin s conjec
ture ab out the holomorphy of Artin L functions is also a consequence of Selb erg s
conjectures
Prop osition If F S then F has no zeros on
Proof Clearly it su ces to prove the assertion for a primitive F The assertion
is true for so we may assume that F is entire Then F s i is also a prim
itive member of S Applying the orthogonality relations to F and we see that
P
a
p
O
1+i
p x p
J B CONREY A GHOSH
The class S
In all known examples of F S it is the case that one may nd a in which all
F
w With this normalization Q is uniquely determined as are the Also
i i
is ambiguous only as far as a factor of Thus is uniquely determined We
are led to consider the p ossibly smaller class of functions S de ned by the same
axioms as S except that the functional equation has the form
d d
Y Y
s s
F s s s F s Q Q
i i
i i
We note that to each member of S there is a unique tuple
d q
d
where
d
q Q
In practice q is always a p ositive integer though we do not make that assumption
We use the notation S d to denote all the elements of S with a given value of d
Thus S consists of all Dirichlet series in S with d
Theorem Supp ose that F S Then q is a p ositive integer and there
exists a primitive Dirichlet character mo d q and a real number such that
F s L s i
Our pro of is similar to Siegel s pro of see or of Hamburger s theorem
though it is formulated somewhat di erently Also Gerardin and Li have given
an adelic pro of of a similar theorem Note the imp ortant role played by the b ound
of axiom of the de nition of the Selb erg class If not for that condition
the Dirichlet series
X X Y
s s s s
n n p
p
n o dd n mo d
would b e an element of S even though this function do es not satisfy the Riemann
Hyp othesis Thus features into our pro of in a signi cant way
Proof We assume that F has a functional equation
s s
s s
Q F s Q F s
If is not real then we consider
G s F s
Then G satis es the functional equation
s s
s s
G s Q G s Q
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
where
Q
Also G s has a p ole p ossibly at s Thus without loss of generality
we may assume that is real and that the p ole of F if it exists is at s i
for some real
Let q Q Then we consider
Z
X
nx
s nx q
f x a x q s F s ds e
n
q i
n
We move the line of integration to the left of and get a residue from the p ole at
s i We make the change of variables s s in the integral and apply
the functional equation Thus
P log x
f x
i
x
Z
s
s s
x q s Q F s ds
i s
We make the change of variables s s and use the duplication formula on the
rst gamma function in the integral Then the integral is
Z
s s
s
x F s ds Q
s
Z
s s s Q
s
F s x
s
x
Z
C
q
s
F s s s x
x
where
Q
C
q
Now we move the line of integration to the right of crossing the p ole at s
i Then we are in a region where the Dirichlet series F s converges
absolutely We expand F s into its Dirichlet series and integrate term by term
We have
X
a n P log x
n
i
f x x P log x C x
q
i
x n x
n
The latter formula comes from the integral formula
Z
b
a b x
s
s b a s x ds
a b
i x
b a
J B CONREY A GHOSH
We divide our equation by x and obtain
X
a n
n
P er x L x C x
x n
n
where P er x is a function that is regular in x and is p erio dic with p erio d iq
Also L x is a function which is regular in the whole plane with the negative real
axis removed and C is indep endent of x Now as x in in the right
side is asymptotic to
a C
n
for some C which is indep endent of n By p erio dicity we have the same asymptotics
as x i n q i n q Therefore q must b e an integer and a a
n n q
Now we use the fact that a is multiplicative It is not di cult to show that if
n
a is a multiplicative function which is p erio dic mo d q then there exists a Dirichlet
n
character mo d q for which a n for all n for which n q It su ces
n
to show that a is completely multiplicative on such n So supp ose that mn q
n
Let r b e an integer for which m r q n Then
a a a a a a
m n m r q n mn
m r q n
whence a is completely multiplicative on those n with n q Now let b e the
n
primitive character which induces Supp ose that is a character with mo dulus
q Then L s satis es a functional equation
r
s
q
s a L s s s
If we form the quotient of the functional equations for F s and L s we obtain
an equation
s s
s
E s s E s CQ
s a s a
for some C which is indep endent of s and where
Y
E s F s L s F s
p
pjq
1
Also a is either or We show that such an equation can only hold if a
and E s
To do this we rst show that all zeros and p oles of s are in For
the quotient of the gamma functions all zeros and p oles are in That E s
has no zeros in follows from the fact that its Euler pro duct involves only
a nite number of factors each of which has no zeros in by Lemma
By the symmetry of the functional equation it follows that all zeros and p oles of
s are on But then the quotient of the gamma functions is entire with
no zeros Hence the gamma functions are the same i e a Then we are
left with a degree functional equation which we ve seen in Theorem implies
that the b are to o large Hence s and F s s or F s L s for
n
a primitive character
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
Functional equations from S We now prove a general converse the
orem ab out Dirichlet series which have GL type functional equations This
theorem may b e regarded as a generalization of the basic theorems of Hecke and
Maass and contains them as sp ecial cases We set up some notation for this
section only Let
X
a
n
F s
s
n
n
b e absolutely convergent for Let
p s s
s
q s F s
We assume that is real as this presents no loss of generality Let J and K denote
the usual Bessel functions and de ne
H x x J x
For x and y p ositive let
X
p p
q K ny q f x y y a H nx
n
n
Theorem s is entire and satis es
s s
if and only if
i
i
f r e f r e
where r and
i
r e x iy
We remark that H x cos x and H x sin x In fact
in the ab ove theorem corresp onds to the case of even Maass forms while
corresp onds to the situation of o dd Maass forms The work of Epstein
Hafner and Sarnak already contains this case Our theorem may b e regarded as
a generalization of their result When we are in the situation of holomorphic
y
cusp forms To see this we rst observe that y K y e Next we
note that if is half an o dd integer say k where k is an even integer
then H x has a zero at x of order k Then di erentiating k times with
resp ect to x and setting x we obtain the usual representation for the mo dular
form On the other hand when we di erentiate k times the relation
y x
f x y f
x y x y
J B CONREY A GHOSH
with resp ect to x and set x and use the fact that the derivatives lower than
k vanish at x we obtain that for some constant c
k
X
k ny
f x y j a n e g y c
x n
x
n
satis es
k
g y y g y
which is the desired relation As an example consider the case of the function
generated by the Ramanujan co e cients n
X
z n e nz
n
This corresp onds to the situation We have
X
n
ny
f x y x J nx e
n
n
and
X
n
s
n F s
n
n
and
x y
f x y f
x y x y
if and only if after use of the duplication formula
s
s F s s s
The latter is of course the well known functional equation after a shift by
for the L function asso ciated with The former is less recognizable One calculates
that
x J x cos x sin x
x x x x x
and that this function has a zero of order at x After di erentiating times
with resp ect to x and setting x one obtains the usual transformation formula
for the function
iy y iy
We remark that f x y ab ove satis es the di erential equation
f x y f x y f x y
xx y y
x y
ON THE SELBERG CLASS OF DIRICHLET SERIES SMALL DEGREES
Also the ab ove theorem remains valid if we replace the Dirichlet series by a gener
alized Dirichlet series
X
s
F s a
n
n
n
and replace the Bessel series by
X
p p
q K y q a J x f x y xy
n n n
n
for a fairly general sequence
n
We b egin with the following lemma ab out the Mellin transform of a pro duct of
Bessel functions
Lemma Supp ose that a and b are real and p ositive Then for s we have
Z
du a
s s s
J au K bu u b
u b
s s
s s a
F
b
Proof of Lemma This lemma follows easily from the transforms
Z
s s dx
s s
J x x
x
and
Z
dx s s
s s
K x x
x
and
Z
s l s m s l m
s
x ds F l m n x
i n s n
c
for s by using the convolution prop erty of Mellin transforms
Z Z
dx
s
f x g x x F z G s z dz
x i
c
where f and F are a Mellin transform pair as are g and G
Lemma Let
s s
s
cot T s sin F
for Then T s T s
Proof of Lemma This fact follows easily from the transformation formula
c a b
F a b c x x F c a c b c x
J B CONREY A GHOSH
and from the fact that F a b c x F b a c x
Proof of Theorem We consider the Mellin transform
Z
dr
i s
f r e r M s
r
Replacing r by r in the integral we see that
i
i
f r e f r e
implies that M s M s By the de nition of f we nd that
X
a n M s sin cos
n
n
Z
p p dr
s
J nr cos q K nr cos q r
r
We evaluate the integral with the help of Lemma and nd that
M s cos cot T s s
s if and only if Since is real it follows from Lemma that s
M s But by Mellin inversion M s
Z
i s
f r e M s r ds
i
c
i
so that replacing s by s we nd that M s M s implies that f r e
i
f r e This completes the pro of
Acknowledgments The rst author would like to thank the Rutgers Mathemat
ics Department for providing an excellent place to work during his sabbatical leave
when this manuscript was prepared He esp ecially thanks Bill Duke and Henryk
Iwaniec for their hospitality during his stay and the many stimulating conversations
regarding this work
Both authors also thank the Institute for Advanced Study for providing a stim
ulating work environment during their so journ to New Jersey
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Department of Mathematics Oklahoma State University Stillwater Oklahoma