Upper and lower bounds at s = 1 for certain Dirichlet series with
Euler product
Giuseppe Molteni
Dipartimento di Matematica
Università di Milano
Via Saldini 50
20133 Milano
ITALY
e-mail: [email protected]
Duke Math. Journal 111(1), 133–158 (2002).
Abstract
Estimates of the form L(j) (s, A) ,j,DA RA in the range |s − 1| 1/ log RA for general
L-functions, where RA is a parameter related to the functional equation of L(s, A), can be
quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates
when the L-functions have Euler product of polynomial type and the Ramanujan hypothesis
is replaced by a much weaker assumption about the growth of certain elementary symmetrical
functions. As a consequence, we obtain an upper bound of this type for every L(s, π), where
π is an automorphic cusp form on GL(d, AK ). We employ these results to obtain Siegel-type
lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass
form.
Mathematics Subject Classification (2000): 11M41
1
Definitions and results
We consider the class of functions satisfying the following axioms:
(A1) (Euler product) Let A = {Ap }p , p prime, be a sequence of complex square matrices of order
d, with monic characteristic polynomial Pp (x) = PpA (x) ∈ C[x] and roots αj (p) = αjA (p).
We define the general L-function L(s, A) as
L(s, A) =
d
YY
(1 − αj (p)p−s )−1 =
p j=1
∞
X
an n−s
n=1
and we suppose the series absolutely convergent for σ > 1.
(A2) (Continuation) There exists m = m(A) ∈ N such that (s − 1)m L(s, A) has entire continuation over C as a function of finite order.
1
(A3) (Growth) There exists 0 < δ < 1/2 such that L(s, A) exp exp(|t|) ∀ > 0 as |t| → ∞,
uniformly in −δ < σ < 1 + δ.
(A4) (Functional equation) There exists a second sequence of complex square matrices A∗ =
(A∗p )p satisfying axioms (A1), (A2), (A3), and there exist QA , λ1 , . . . , λr ∈ R+ , µ1 , . . . , µr ∈
C with <µj ≥ 0 ∀j and ω ∈ C with |ω| = 1 such that the functions
Φ(s, A) = QsA
r
Y
Φ(s, A∗ ) = QsA
Γ(λj s + µj )L(s, A),
j=1
r
Y
Γ(λj s + µ̄j )L(s, A∗ )
j=1
satisfy the functional equation
Φ(1 − s, A) = ωΦ(s, A∗ ).
By definition, the main parameter of L(s, A) is the quantity
RA = (1 + QA )
r
Y
(1 + |µj |).
j=1
(A5) (Tensor product) There exists a finite, possibly empty, set of primes PA , the exceptional
set, and complex numbers γi (p), δi (p) ∈ C for i = 1, . . . , d2 , p ∈ PA with
|γi (p)|, |δi (p)| ≤ p1−ρ
for some ρ > 0, ∀i = 1, . . . , d2 , p ∈ PA ,
such that, if we define
L(s, A ⊗ Ā) =
d
Y Y
(1 − αi (p)ᾱj (p)p−s )−1 ,
p i,j=1
2
P (s, A ⊗ Ā) =
d
Y Y
(1 − γi (p)p−s )(1 − δi (p)p−s )−1 ,
p∈PA i=1
^
L(s, A
⊗ Ā) =P (s, A ⊗ Ā)L(s, A ⊗ Ā),
^
then L(s, A
⊗ Ā) satisfies axioms (A1)-(A4). By abuse of notation we denote by RA⊗Ā
^
the main parameter of L(s, A
⊗ Ā) and we assume that
X
1 log RA⊗Ā ,
(1)
p∈PA
log RA⊗Ā log RA .
(2)
It is important to remark that we do not assume anything about the size of the eigenvalues
αj ; in particular, the Ramanujan hypothesis is not assumed. This is the only difference with
a class of functions introduced and widely studied in Carletti-Monti Bragadin-Perelli [3], so we
refer to that paper for the basic properties of the functions L(s, A) and for comments about the
axioms; here we only recall that the exceptional set PA of axiom (A5) is related to the existence
of ramified primes.
2
Remark. For well known classes of L-functions the main parameter RA captures, although in
a quite rough form, most of the algebraic informations about L(s, A): for examples, for the
Dirichlet L(s, χ)-functions, R q, i.e., the modulus of χ; for Hecke L-functions related to
holomorphic cuspidal forms, R kN , i.e., the product of the level N and the weight k; for
Maass L-functions, R λN , i.e., the product of the level N and the eigenvalue λ; and in
general, for the L-functions associated with cuspidal automorphic representations of the groups
GL(d), R is of the order of the analytic conductor introduced by Iwaniec and Sarnak in [13]. For
this reason our results will be uniform in the R-aspect but will depend on all other parameters
appearing in axioms. It is useful, therefore, to introduce the following notation: let DA be the
set
DA = {mA , mA⊗Ā , rA , rA⊗Ā , dA , λj (A), λj (A ⊗ Ā)}
i.e., DA contains the order of pole mA , the number of gamma factors rA , the degree dA and
the λj coefficients, both of L(s, A) and of L(s, A ⊗ Ā). In this way we will say our results are
RA -independent and DA -dependent.
The aim of this paper is proving an upper bound for L(1, A), and more generally for
of type
L(j) (s, A),
L(j) (s, A) ,j,DA RA
for
|s − 1| 1/ log RA .
(3)
Such estimates are an essential tool for a general Siegel-type estimate, as we will see in Section 2.
Obtaining these estimates is quite easy when the Ramanujan hypothesis is assumed (see [3]).
However, this hypothesis has been proved only for a limited class of functions (the Hecke Lfunctions, the Artin L-functions and the L-functions coming from the cuspidal holomorphic
forms for congruence groups, see Deligne [4]), although it is generally believed that all the Lfunctions appearing in number theory should satisfy the Ramanujan hypothesis; for example, it
is conjectured to hold for the L-functions associated with cuspidal automorphic representations.
In general, only partial and rather poor estimates for the coefficients are at our disposal, hence
it is interesting to consider the possibility to obtain (3) without the Ramanujan hypothesis.
For Maass forms (d = 2) this has been done firstly by Iwaniec [12] (in the Q-aspect, in
P
that paper). He remarks that a preliminary estimate for S(x) =
n≤x |an |/n of the form
S(x) Qc x for some constant c and for every > 0 can be proved in a standard way. Then,
the multiplicative properties of the coefficients of these functions can be employed to obtain
an estimate for S(x)2 in terms of S(x2 ). By iterating this relation the fundamental estimate
S(x) (Qx) is deduced, and the claim easily follows. This method has been also used
by Hoffstein-Lockhart [10] to get an analogous estimate in the case of the symmetric square
L-functions (d = 3) associated with Maass forms.
We modify Iwaniec’s idea in such a way to obtain the required estimate for functions of
any degree d; in this sense the most original part of our work is Section 2, where the basic
Proposition 1 below is proved.
We introduce the following notation: sj (p) denotes the j-th elementary symmetric function
of the roots α1 (p), . . . , αd (p), i.e.,
sj (p) =
X
αi1 (p) · · · αij (p);
1≤i1 <···<ij ≤d
3
moreover, let R(p) =
d
√ X
2
|sj (p)|1/j and let R(d) be the completely multiplicative function
j=2
generated by R(p).
Proposition 1. We have
|an am | ≤
X∗
R(d)|a nm
|
d
∀n, m ≥ 1,
(4)
d|(n,m)d
d|nm
P∗
where
denotes that the summation is restricted to square-full divisors d, i.e., either d = 1 or
d > 1 and p2 |d for every p|d.
Remark. When d = 1 the estimate (4) is reduced to the trivial |an am | ≤ |anm |.
Obviously, some hypothesis about the size of the coefficients has to be assumed in order to
prove (3). We assume
Hypothesis (R). There exists ρ > 0 such that |αj (p)| ≤ p1−ρ for every prime p, and the
estimate
sj (p) pj/2
∀ 2 ≤ j ≤ d , uniformly over R
holds.
Remark. The first part of the hypothesis means that the local components are convergent when
σ > 1 − ρ. Moreover, the estimate on the elementary symmetric functions is satisfied in an
obvious way when the estimate an n1/2 holds, uniformly on R. Nevertheless, hypothesis (R)
does not assume anything about the first symmetric function s1 (p), in this way hypothesis (R)
is satisfied as well in some cases where the global estimate an n1/2 is not known.
At last, it is also interesting to remark that the quality of the estimate assumed in (R) does not
depend on the degree d, as instead a consideration of similar situations could suggest.
Our main result is
Theorem 1. Let L(s, A) satisfy axioms (A1)-(A5) and hypothesis (R) and define f (s) = (s −
1)mA L(s, A). Then
f (j) (s) ,D (j + 1)cj R logj R +
j!
2j R
for
|s − 1| 1/ log R,
(5)
uniformly on j, for a suitable positive constant c = c(D), independent of R.
Remark. Under the same hypotheses and with some minor change to the proof of this theorem
we caught obtain upper bounds of similar type for the values of L(s, A) when s is centered at a
different point of the line σ = 1; for example, for θ 6= 0 we have
L(s + iθ, A) ,D (R(1 + |θ|))
for
|s − 1| 1/ log(R(1 + |θ|)).
(6)
It is interesting to remark that this more general result is already included in Theorem 1 when
L(s, A) is entire. In fact, in this case the shifted L-function L(s, A(θ)) = L(s + iθ, A) satisfies
axioms (A1)-(A5) and (R) with log RA(θ) DA log RA + log(1 + |θ|) so that (5) for L(s, A(θ))
gives (6) for L(s, A).
4
For the applications it is important to know estimate (3) for the generic tensor product
function
L(s, A ⊗ B) =
dA Y
dB
YY
(1 − αi (p)βj (p)p−s )−1
p i=1 j=1
as well; to deal with this function, axiom (A5) has to be modified in the following way.
(A5’) There exists a finite, possibly empty, set of primes PA,B and complex numbers γi (p), δi (p) ∈
C for i = 1, . . . , dA dB , p ∈ PA,B with
|γi (p)|, |δi (p)| ≤ p1−ρ
for some ρ > 0, ∀i = 1, . . . , dA dB , p ∈ PA,B ,
such that, if we define
P (s, A ⊗ B) =
Y
dY
A dB
(1 − γi (p)p−s )(1 − δi (p)p−s )−1
p∈PA,B i=1
^
L(s, A
⊗ B) =P (s, A ⊗ B)L(s, A ⊗ B),
^
then L(s, A
⊗ B) satisfies axioms (A1)-(A4). As before, by abuse of notation we denote
^
by RA⊗B the main parameter of L(s, A
⊗ B) and we assume that
X
1 log RA⊗B ,
(7)
log RA⊗B log RA + log RB .
(8)
p∈PA,B
Remark. Under hypotheses (7) and (8), the estimate
(RA RB )− P (1, A ⊗ B) (RA RB )
^
holds, so that any upper bound of type (3) for L(s, A
⊗ B) gives a similar upper bound for
L(s, A ⊗ B) and vice versa. Moreover, in the context of L-functions from representations of
^
GL(d), L(s, A
⊗ B) is the Rankin-Selberg convolution and estimates (1), (2), (7) and (8) about
the ramified primes are satisfied.
^
An upper bound for L(s, A
⊗ B) can be deduced by Theorem 1, but in this case we have to
^
assume axiom (A5) for L(s, A ⊗ B), i.e., the validity of (A1)-(A4) for L(s, A ⊗ Ā ⊗ B ⊗ B̄). This
fact agrees with well known conjectures, but it is not proved in general, hence it is important
for applications to obtain an upper bound for L(s, A ⊗ B) under a different set of hypotheses.
The following Theorems 2 and 3 achieve this purpose.
Theorem 2. Assume that both L(s, A) and L(s, B) satisfy axioms (A1)-(A5) and (A5’).
MoreQ
dA
over, assume that both L(s, A ⊗ Ā) and L(s, B ⊗ B̄) satisfy hypothesis (R) and that i=1 αi (p),
Q
dB
j=1 βj (p) ≤ 1 for every p. Define f (s) = (s − 1)mA⊗B L(s, A ⊗ B). Then
f (j) (s) ,DA ,DB (j + 1)cj (RA RB ) logj (RA RB ) +
j!
2j R
A RB
for
|s − 1| 1/ log(RA RB ),
uniformly on j, for a suitable positive constant c = c(DA , DB ), independent of RA , RB .
5
Nevertheless, there are cases where (R) is not proved for L(s, A ⊗ Ā), but the following
stronger hypothesis about L(s, A) holds.
Hypothesis (R’). There exists ρ > 0 such that |αj (p)| ≤ p1−ρ for every prime p and the
estimate
sj (p) pj/4
∀ 2 ≤ j ≤ d , uniformly over R
holds.
We can prove upper bounds for L(s, A ⊗ B) under this hypothesis as well.
Theorem 3. Assume that both L(s, A) and L(s, B) satisfy axioms (A1)-(A5) and
over, assume that both L(s, A) and L(s, B) satisfy hypothesis (R’) and that
(A5’).
MoreQ
dA
i=1 αi (p),
Q
dB
j=1 βj (p) ≤ 1 for every p. Then, defining f (s) = (s − 1)mA⊗B L(s, A ⊗ B) we have
f (j) (s) ,DA ,DB (j + 1)cj (RA RB ) logj (RA RB ) +
j!
2j R
A RB
for
|s − 1| 1/ log(RA RB ),
(9)
uniformly on j, for a suitable positive constant c = c(DA , DB ), independent of RA , RB .
According to the original Iwaniec’s idea, Theorems 1-3 are deduced in a standard way from
suitable upper bound for the coefficients of L(s, A) and L(s, A ⊗ B); we state here explicitly the
upper bound giving Theorem 1 since it has an interest of its own; the similar upper bounds for
the coefficients of L(s, A ⊗ B) under the hypotheses of Theorems 2 and 3 are contained in their
proofs.
Theorem 4. Let L(s, A) satisfy axioms (A1)-(A5) and hypothesis (R). Then
X |an |
n≤x
n
,D (Rx)
∀ > 0.
(10)
Examples. Let α(p) and α−1 (p) be the coefficients of the Euler product of the L-function
L(s, f ) associated with a Maass form f . Then d = 2 and s2 (p) = 1 so (R) is satisfied for
L(s, f ), hence for this function estimate (5) holds, as already proved by Iwaniec [12].
Moreover, Kim and Shahidi [15] proved that p−5/34 < |α(p)| < p5/34 . The square-symmetric
L-function L(s, sym2 f ) associated with f satisfies axioms (A1)-(A5) by the works of Shimura
[20], Gelbart-Jacquet [5] and Moeglin-Waldspurger [17]. It has an Euler product of degree 3
and α2 (p), 1, α−2 (p) as coefficients, thus s3 (p) = 1 and |s2 (p)| = |α2 (p) + 1 + α−2 (p)| ≤ 3p5/17 ,
hence (R) is satisfied and the estimate L(1, sym2 f ) R holds by Theorem 1. This has been
proved already by Hoffstein-Lockhart [10].
It is easy to verify that the estimate of Kim and Shahidi for the coefficients of a Maass form
is sufficiently strong to prove that L(s, sym2 f ⊗ sym2 f ) satisfies (R), so that by Theorem 2 it
follows that ress=1 L(s, sym2 f ⊗ sym2 f ) R , again a result already proved in [10] (in that
paper the upper bound for ress=1 L(s, sym2 f ⊗ sym2 f ) was not deduced by Theorem 2 but by a
specific version of Theorem 3 because only the weaker estimate p−1/5 < α(p) < p1/5 was known
at that time).
At last, Maass functions are examples of Langlands L-functions, a general and extremely
important class of functions. This theory associates an L-function, L(s, π), to every automorphic
6
cuspidal representation π of GL(d, AK ), where AK is the Adèle ring of a global field K (see [6]).
These functions satisfy axioms (A1)-(A5) by the work of many authors (see [7], [19] and [17]);
by Luo-Rudnick-Sarnak [16] the local coefficients of these functions satisfy
2 +1)
|αj (p)| < p1/2−1/(d
for any j = 1, . . . , d,
hence all the hypotheses of Theorem 1 are satisfied (but it is interesting to remark that the
old result of Jacquet and Shalika [14] asserting the bound |αj (p)| < p1/2 is sufficient for this
purpose) so that Corollary below immediately follows.
Corollary. Let π be an automorphic cuspidal representation of GL(d, AK ) and L(s, π) be the
associate L-function. Let f (s) = (s − 1)m L(s, π), where m is the order of pole of L(s, π) at
s = 1, then
|f (j) (s)| ≤ c R for |s − 1| 1/ log R,
for a suitable c = c(, j, m, d) > 0.
(For the sake of simplicity, we ignore here uniformity on the order j.) An unconditional and
general statement corresponding to Theorem 2 or 3 is not possible for the Rankin-Selberg convolution L(s, π ⊗ π 0 ) since it is not known if this function satisfies axiom (A5) or if L(s, π) satisfies
Hypothesis (R’).
Siegel-type lower bounds
Upper bounds of Theorems 1-3 are necessary ingredients for the proof of Siegel-type theorems,
i.e., for the proof of lower bounds of the form
L(1, A) ,D R−
∀ > 0;
(11)
the importance of such a type of results justifies the previous researches about lower bounds
for L-functions whose coefficients are not known to satisfy Ramanujan hypothesis. A careful
analysis about Siegel-type lower bounds is carried out in [18], where the results of GolubevaFomenko [8, 9] about a lower bound for holomorphic cusp forms, the well known estimate (11)
for symmetric square power of a Maass form proved by Hoffstein-Lockhart [10] and similar
results are deduced as consequences of a coherent and axiomatic approach. In [18] new results
are deduced as well; in the following we show how some of these results, i.e., estimates (12)-(14)
below, can be proved.
Let f ∈ S0 (Γ0 (N ), ψ), i.e., let f be a Maass cusp form for the Hecke congruence subgroup
of level N with the real character ψ modulus N as multiplier. As we have shown in previous
examples, both L(s, f ) and L(s, sym2 f ) satisfy (R’). Moreover, let χ and χ0 be different real
primitive and non-principal characters modulo q and q 0 respectively. It is known that Lχ (s, f ) =
L(s, χ ⊗ f ), Lχ (s, sym2 f ) = L(s, χ ⊗ sym2 f ) and Lχ (s, f ⊗ sym2 f ) = L(s, fχ ⊗ sym2 f ) are
entire functions satisfying axioms (A1)-(A5) (by [17]) and that
log Rχ⊗f log Rf + log q
7
and the same is true for Rχ⊗sym2 f and Rfχ ⊗sym2 f . We consider
F (s) =L s, (1 + χ) ⊗ (1 + χ0 ) ⊗ (f + sym2 f ) ⊗ (f + sym2 f )
=ζ(s)L(s, sym2 f )L(s, sym2 f ⊗ sym2 f )L2 (s, f ⊗ sym2 f )·
L(s, χ)Lχ (s, sym2 f )Lχ (s, sym2 f ⊗ sym2 f )L2χ (s, f ⊗ sym2 f )·
L(s, χ0 )Lχ0 (s, sym2 f )Lχ0 (s, sym2 f ⊗ sym2 f )L2χ0 (s, f ⊗ sym2 f )·
L(s, χχ0 )Lχχ0 (s, sym2 f )Lχχ0 (s, sym2 f ⊗ sym2 f )L2χχ0 (s, f ⊗ sym2 f );
this function has a representation as a Dirichlet series with positive coefficients, as it can be
verified by considering its logarithmic derivative. It has a double pole at s = 1 and is divisible
by Lχ (s, f ⊗ sym2 f )Lχ0 (s, f ⊗ sym2 f ) with multiplicity two. By Theorems 1-3 upper bounds
of the form f, (qq 0 ) follow for the derivatives of every order of any function appearing into
F (s). Therefore, by the standard approach to Siegel-type theorems (see [8], for example), we
prove that
Lχ (1, f ⊗ sym2 f ) f, q −
∀ > 0.
Since Lχ (s, f ⊗sym2 f ) = L(s, f )Lχ (s, sym3 f ), where L(s, sym3 f ) is the L-function investigated
by Kim and Shahidi [15], we deduce that
Lχ (1, sym3 f ) f, q −
∀ > 0.
(12)
In a similar way we can prove that
Lχ (1, f ⊗ sym2 g) f,g, q − ,
2
−
2
Lχ (1, sym f ⊗ sym g) f,g, q ,
(13)
(14)
for f and g both Maass forms, f 6= g.
We recall that βA , the largest real zero of L(s, A), is called Siegel zero of L(s, A) if 1 − βA 1/ log RA , and that (11) is equivalent to
1 − βA ,DA R−
A .
Banks [1] and Hoffstein-Ramakrishnan [11] have proved that L-functions related to automorphic
cuspidal representations π of GL(2) and GL(3) have not Siegel-zero, so that for these functions
the stronger estimate
1
L(1, π) log R
holds. However, estimates (12)-(14), involving functions conjecturally related to automorphic
representations of GL(4), GL(6) and GL(9), are non-trivial.
2
2.1
Algebraic relations
Local relations
Let {bh }∞
h=0 be a sequence with b0 = 1, extended to h < 0 by setting bh = 0 in this range. We
assume that the sequence bh verifies the relation
b1 bh = bh+1 +
d
X
u=2
8
xuu bh+1−u ,
(15)
where d ≥ 1 is a fixed integer, the degree of the sequence, and {xu }du=2 are non-negative real
parameters.
Remark. When d = 1 the relation is reduced to b1 bh = bh+1 .
With these assumptions we prove
Proposition 2. There exist polynomials Pu,l ∈ N[x2 , · · · , xd ] such that
dl
X
bl bh =
∀l, h ≥ 0.
Pu,l (x) bh+l−u
(16)
u=0
Moreover, if we assume P0,0 = 1 and Pu,l = 0 in the following cases: u < 0 or l < 0 or u > dl
or u = 1, such polynomials satisfy the recursion
Pu,l+1 (x) = Pu,l (x) +
d
X
xgg Pu−g,l (x) − Pu−g,l+1−g (x) .
(17)
g=2
Proof. We introduce the variables x0 and x1 , since in this way we can write b1 bh = du=0 xuu bh+1−u
and recover relation (15) by setting x1 = 0. We prove the claim recursively on l. By (15), it is
true when l = 1, for every h. Assume now the claim for the values ≤ l and for every h, we will
verify it for l + 1, for every h again. We have
P
dl
X
b1 (bl bh ) =
Pu,l (x) b1 bh+l−u =
Pu,l (x)
dl X
d
X
d
X
xgg bh+l+1−u−g
g=0
u=0
u=0
=
dl
X
d(l+1)
xgg Pu,l (x) bh+l+1−u−g =
u=0 g=0
= bh+l+1 +
d(l+1)
X
X
γ=0
g+u=γ
0≤g≤d
xgg Pu,l (x) bh+l+1−γ
X
X
γ=1
g+u=γ
0≤g≤d
xgg Pu,l (x) bh+l+1−γ ,
moreover,
d
X
(b1 bl )bh =
xuu bl+1−u bh = bl+1 bh +
d
X
xuu bl+1−u bh
u=1
u=0
= bl+1 bh +
d d(l+1−u)
X
X
u=1
= bl+1 bh +
ρ=0
dl X
γ=1
xuu Pρ,l+1−u (x) bh+l+1−ρ−u
X
xuu Pρ,l+1−u (x) bh+l+1−γ .
u+ρ=γ
1≤u≤d
0≤ρ≤d(l+1−u)
Comparing these identities we get
bl+1 bh = bh+l+1 +
d(l+1)
X
X
γ=1
g+u=γ
0≤g≤d
xgg Pu,l (x) −
h
X
g+u=γ
1≤g≤d
0≤u≤d(l+1−g)
9
i
xgg Pu,l+1−g (x) δγ≤dl bh+l+1−γ ,
(18)
where δγ≤dl is 1 or 0 according to γ ≤ dl or otherwise. The existence of the polynomials Pu,l+1
follows from (18), completing in this way the proof of (16). Moreover,
Pγ,l+1 (x) =
X
xgg Pu,l (x) −
g+u=γ
0≤g≤d
h
i
X
xgg Pu,l+1−g (x) δγ≤dl
g+u=γ
1≤g≤d
0≤u≤d(l+1−g)
= Pγ,l (x) +
X
xgg Pu,l (x) −
h
X
i
xgg Pu,l+1−g (x) δγ≤dl , (19)
g+u=γ
2≤g≤d
0≤u≤d(l+1−g)
g+u=γ
2≤g≤d
where the assumption x1 = 0 has been introduced. Since Pu,l = 0 for u < 0, or d < 0 or u > dl,
(19) implies (17).
The particular form of (17) suggests the analysis of the polynomials Dl,h = Pu,l − Pu,l−1 ; we
get
Proposition 3. The polynomial sequence Du,l = Pu,l − Pu,l−1 verifies the recursion

Pd
g Pg−1


Du,l+1 (x) = g=2 xg ρ=1 Du−g,l+1−ρ (x)
D
=1
0,0


D = 0 when u < 0, or u > dl, or l < 0.
u,l
(20)
Moreover, Du,l ∈ N[x2 , · · · , xd ] and it is homogeneous of degree u, i.e., Du,l (λx) = λu Du,l (x)
for every λ and x.
Proof. The recursive relation (20) is deduced from (17). The other claim is an easy consequence
of (20).
We need another property of the sequence Du,l .
Proposition 4. If l ≥ u + d, then Du,l = 0.
Proof. We prove the claim recursively on u; it is true when u = 0 and u = 1 for every l > 0. We
suppose the claim to hold up to u, and we prove it for u + 1. Let l ≥ u + 1 + d. By the identity
Du+1,l =
d
X
g=2
xgg
g−1
X
Du+1−g,l−ρ ,
ρ=1
the claim is the same as proving that every Du+1−g,l−ρ is zero. The inductive hypothesis assures
it if l − ρ ≥ u + 1 − g + d, and this inequality holds since we assumed l ≥ u + 1 + d and ρ < g.
Remark. Let l0 (u) be the smallest value such that Du,l = 0 when l ≥ l0 (u). Proposition 4 states
that l0 (u) ≤ u + d, but it is easy verify that the inequality is not sharp. Nevertheless, the exact
value of l0 (u) is not necessary here.
10
Every Du,l (x) ∈ N[x], therefore Pu,l1 (x) ≤ Pu,l2 (x) when l1 ≤ l2 (recall we assume xi ∈ R+ )
P
since Pu,l = lg=0 Du,g ; nevertheless, the sequence Pu,l is bounded on l since by Proposition 4
the following series is finite
Pu (x) =
∞
X
Du,g (x)
g=0
so that
Pu,l (x) ≤ Pu (x)
∀ xi ∈ R+ .
∀u, l,
(21)
The existence of the polynomials Pu giving an upper bound for Pu,l , uniform on l, is an essential
fact; we summarize their principal properties in the following proposition.
Proposition 5. The sequence Pu (x) satisfies the recurrence

Pd
g


Pu (x) = g=2 (g − 1)xg Pu−g (x)
P =0
u


P = 1, P = 0.
0
1
if u ≥ 2
if u < 0
(22)
Moreover, Pu belongs to N[x2 , · · · , xd ], is homogeneous with degree u and satisfies the estimate
√
(23)
Pu (x) ≤ [ 2(x2 + · · · + xd )]u .
Proof. By definition Pu = ∞
l=0 Du,l , therefore Pu belongs to N[x] and is homogeneous with
degree u because the same properties hold for every Du,l . The recursive relation (22) is recovered
from (20). In order to prove (23) we first remark that if set
P
fM (y) =
M h
X
yi
ii
PM
j=1 yj
i=1
,
then fM (y) ≤ 1 when yi ∈ R+ for every i; we prove this fact inductively over M . It is true for
M = 1. By the inductive hypothesis fM −1 (y1 , . . . , yM −1 ) ≤ 1, we get
y1
1 ≥ fM −1 (y1 + yM , y2 , . . . , yM −1 ) = PM
j=1 yj
y1
≥ PM
j=1 yj
h y
M
+ PM
iM
j=1 yj
+
M
−1h
X
i=2
yi
yM
+ PM
ii
PM
j=1 yj
j=1 yj
+
M
−1h
X
i=2
yi
ii
PM
j=1 yj
= fM (y1 , y2 , . . . , yM ).
Now we can prove (23) once again by induction. It is true when u = 0 and it is trivial when
u < 0. By the recursive law (22), we have
Pu (x) =
d
X
(g − 1)xgg Pu−g (x) ≤
d
hX
d
(g − 1)xg i √ X
√ Pd g
( 2
xj )u
g
(
2
x
)
j=2 j
g=2
j=2
g=2
≤
d
hX
g=2
xgg
Pd
(
g
j=2 xj )
d
i √ X
( 2
where we have used the inequality g − 1 ≤
d
√ X
xj ) ≤ ( 2
xj )u ,
j=2
2g/2
u
j=2
and the previous estimate fM (y) ≤ 1.
Remark. Since every Du,l is homogeneous, Pu,l is homogeneous too (as directly verified by (17)).
This immediately implies that there exists a constant c = c(l) such that Pu,l (x) ≤ (c(l)(x2 +· · ·+
xd ))u , for every u. Obtaining (23) is important since it shows that c(l) is actually independent
of l.
11
2.2
Euler product: proof of Proposition 1
Now we use the results of the previous section for the study of the coefficients of an Euler
product. Let {αj }dj=1 be complex numbers, and let {bh }∞
h=0 be the sequence generated by the
relation
∞
X
bh xh =
d
Y
(1 − αj x)−1 .
j=1
h=0
Let {sj }dj=1 be the elementary symmetric polynomials of the variables αj , then the sequence bh
satisfies the recursion
(
bh − s1 bh−1 + s2 bh−2 + · · · + (−1)d sd bh−d = 0
b0 = 1.
∀h > 0
But s1 = b1 , so the recursive relation can be formulated as
b1 bh−1 = bh + s2 bh−2 + · · · + (−1)d sd bh−d .
Comparing this relation with (15), by (16), (21) and (23) we get
d min {l,h}
|bl bh | ≤ |bl+h | +
X
u
R |bl+h−u |,
with
d
√ X
|sj |1/j .
R= 2
u=2
j=2
Proposition 1 follows from this estimate by multiplicativity.
2.3
A useful proposition
For the proof of the theorems we will need the following lemma establishing an inequality between
the coefficients of the series L(s, A) and those of the convolution series L(s, A ⊗ Ā).
Proposition 6. Let L(s, A) =
for every n.
P∞
n=1 an n
−s
and L(s, A ⊗ Ā) =
P∞
n=1 An n
−s .
Then An ≥ |an |2
2 −s holds when the degree is 2 by a formula
The identity L(s, A ⊗ Ā) = ζ(2s) ∞
n=1 |an | n
of Ramanujan, and in this case the statement easily follows by this fact because ζ(2s) is a
Dirichlet series with non-negative coefficients. For larger degrees the relation L(s, A⊗ Ā) =
P
2 −s again holds for a suitable Dirichlet series F (s) but in this case the coefficients
F (s) ∞
n=1 |an | n
of F (s) can be negative so that the general inequality of Proposition 6 cannot be proved in this
way.
P
Proof. Both L(s, A) and L(s, A ⊗ Ā) have a representation as an Euler product, thus it is
Q
sufficient prove the claim for the local components, i.e., verify that if f (x) = dj=1 (1 − αj x)−1 =
P∞
Q
P
d
∞
h
−1 =
h
2
h=0 ah x and F (x) =
i,j=1 (1 − αi ᾱj x)
h=0 Ah x , then Ah ≥ |ah | . We remark that
∞
X
h=0
hah xh = x
d
d X
∞
∞
∞
X
X
X
X
d
αj x
f (x) = f (x)
= f (x)
αjh xh =
ah xh ·
τh xh ,
dx
1
−
α
x
j
j=1
j=1 h=1
h=0
h=1
12
where we have set τh =
∞
X
hAh xh = x
h=0
Pd
h
j=1 αj ;
in a similar way, for F (x) we have
d
d X
∞
∞
∞
X
X
X
X
αi ᾱj x
d
F (x) = F (x)
= F (x)
αih ᾱjh xh =
Ah xh ·
|τh |2 xh .
dx
1
−
α
ᾱ
x
i
j
i,j=1
i,j=1 h=1
h=0
h=1
Therefore, the following recursions hold
(
Ph
hah =
a0 = 1,
l=1 τl ah−l
Ph
(
∀h > 0
hAh =
A0 = 1.
2
l=1 |τl | Ah−l
∀h > 0
(24)
These relations prove that for every h ≥ 0 there exists a polynomial Ph ∈ N[x1 , · · · , xh ] such
that
h! ah = Ph (τ1 , · · · , τh )
h! Ah = Ph (|τ1 |2 , · · · , |τh |2 ),
therefore, proving Ah ≥ |ah |2 is the same as showing that
h! Ph (|τ1 |2 , · · · , |τh |2 ) ≥ |Ph (τ1 , · · · , τh )|2 .
(25)
Ph (1, · · · , 1) = h!,
(26)
At last, we observe that
since if we set τl = 1 for every l, then the recursion (24) is solved by Ah = 1 for every h.
We know that Ph ∈ N[x1 , · · · , xh ], hence by (26) we can consider Ph as a sum of h! monomials,
every one with unitary coefficient, so that (25) follows from the Cauchy-Schwarz inequality
Ph!
P
2
2
h! h!
i=1 |yi | ≥ | i=1 yi | .
Remark. Proposition 6 also follows from Littlewood’s lemma quoted in Subsection 3.3, but we
believe our proof is conceptually easier.
3
3.1
Proof of Theorems
Theorem 4
Let S be a finite set of primes, independent of R and which we will choose later on. We follow
the notation of the axioms and we consider the quantity
SS (x) =
X
n≤x
(n,S)=1
|an |
.
n
The first step is proving that there exists a constant c1 = c1 (D) > 0, independent of R, such
that
SS (x) ,D Rc1 x ,
∀ > 0.
(27)
13
Proof. By the Stirling asymptotic formula, when λ > 0, <µ ≥ 0, and s = σ + it with σ < 0, we
get
Γ(λ(1 − s) + µ̄)
σ,λ [(1 + |µ|)(1 + |t|)]λ(1−2σ) ,
Γ(λs + µ)
uniformly on t and µ. This upper bound, axioms (A1)-(A4) and the Lindelöf principle imply
that the function L(s, A) has a polynomial growth in the Rt-aspect on every vertical strip, i.e.,
that the estimate
(s − 1)mA L(s, A) D (RA (1 + |t|))c1
(28)
holds when a < σ < b, for some constant c1 = c1 (a, b, D) > 0. By axiom (A5) a similar estimate
^
holds for L(s, A
⊗ Ā), and by assumptions (1) and (2) about the exceptional set PA , the same
estimate holds for L(s, A ⊗ Ā), i.e.,
(s − 1)mA⊗Ā L(s, A ⊗ Ā) D (RA (1 + |t|))c1 .
(29)
Therefore,
X An
n
n≤x
2r r!
n r r
2 =
1−
n
2x
2πi
X An ≤
n≤2x
Z
+ 1, A ⊗ Ā)(2x)s
ds,
s(s + 1) · · · (s + r)
+i∞L(s
−i∞
where r = r(c1 ) is a large parameter assuring the convergence of the integral; by (29) we get
X An
n≤x
n
,D Rc1 x
∀ > 0, for some c1 = c1 (D) > 0.
By Proposition 6 we have |an | ≤ 1 + |an |2 ≤ 1 + An , so that SS (x) ≤
immediately follows from (30).
P
n≤x
(30)
1+An
n ,
and (27)
Iwaniec’s work suggests the next step, that is, showing that
SS2 (x) ≤ c2 x SS (x2 )
for some c2 = c2 (, D) > 0.
(31)
Proof. By Proposition 1 we have
SS2 (x) =
X
n,m≤x
(nm,S)=1
≤
X
n,m≤x
(nm,S)=1
|an am |
nm
1
mn
X∗
R(d)|a nm
|=
d
d≤x2
(d,S)=1
so that
d
A≤x2
(A,S)=1
X∗ R(d) |a nm |
d
n,m≤x d|(n,m)d
(nm,S)=1
d|nm
d|(n,m)d
d|nm
X∗ R(d) X
≤
X
d
nm
d
|aA | max #{(n, m) : n, m < x, nm = AD},
A A,D≤x2
X∗ R(d) SS2 (x) log x
d≤x2
(d,S)=1
14
d
SS (x2 ).
(32)
Now we estimate the sum in (32). Recalling that R is a completely multiplicative function and
P
that ∗ denotes a sum over the square-full divisors only, we get
X∗ R(d)
d≤x2
(d,S)=1
d
≤
Y
1+
p≤x2
p6∈S
∞
Y
R2 (p) R3 (p)
R2 (p) X
R(p) j 1
+
.
+
+
·
·
·
≤
2
3
2
p
p
p j=0 p
2
p≤x
p6∈S
By hypothesis (R) and the definition of R(p) there exists c3 dependent on D but independent
of R such that R(p) ≤ c3 p1/2 for every prime. Hence, the series in the infinite product is always
convergent if a convenient set S is assumed, for example S = {p : p ≤ c23 + 1}. With this choice
of S the estimate becomes
X∗ R(d)
d≤x2
(d,S)=1
d
D
Y
Y
c5 R2 (p) c6 1
+
D logc6 x,
≤
p2
p
2
1+
p≤x2
p6∈S
p≤x
and (31) is proved in the stronger form SS2 (x) D logc6 +1 x SS (x2 ).
By iterating (31) and using (27) we obtain, for every M > 0,
M
SS2 (x) ≤ cM x2
M
M
SS (x2 ) ≤ c Rc1 x2
M
for some c = c(, D, M ) > 0,
hence, taking the 2M -th root with a suitable M = M (c1 , ) = M (, D), we get
SS (x) ,D (Rx) .
(33)
h
Since the local factors are convergent for σ > 1 − ρ by (R), the series ∞
h=0 |aph |/p converges
for every p and is bounded by a constant independent of R. Hence, from (33) we have
P
X |an |
n≤x
3.2
n
≤
∞
X |a | Y X
n
n≤x
(n,S)=1
n
|aph |/ph SS (x) ,D (Rx) .
p∈S h=0
Theorem 1
Theorem 1 is deduced by Theorem 4 in the following way. From (33) we get
X
n≤x
logj n
X |an |
|an |
≤ logj x
,D (Rx) logj x,
n
n
n≤x
uniformly on j. Let I (j) (x) be defined by
I (j) (x) =
1
2πi
Z
2+i∞
2−i∞
f (j) (s + 1)xs
ds
s(s + 1) · · · (s + r)
(34)
with r = r(D), independent of j and sufficiently large to assure the convergence of the integral.
We remark that
dj sm Pj,m (s, log n)
=
dsj ns
ns
15
with Pj,m (x, y) a polynomial bounded by mj+1 (j + 1)xm y j uniformly on m and j. Moreover,
1
2πi
Z
2+i∞
2−i∞
sm
x s
ds =
s(s + 1) · · · (s + r) n
hence,
I (j) (x) mj+1
A (j + 1)
(
0
rm
r!
1+
n r
x
≤
r m 2r
r!
unif. on r
if n > x
if n ≤ x,
rmA 2r X
|an |
logj n
,D (j + 1)cj (Rx) logj x
r! n≤x
n
uniformly on j, for some c = c(D) > 0. Moreover, by (28) and the Cauchy theorem about the
derivative of an holomorphic function, we get
(j)
f (j) (s) = (s − 1)mA L(s, A)
D j! 2−j (R(1 + |t|))c1
uniformly on j, with a < σ < b.
We move the integration line in (34) to σ = −1/2 so that by the residue theorem we obtain
I (j) (x) = f (j) (1)/r! + O
j!Rc1 √ ;
2j x
choosing x = R2c1 +2 we get the estimate
f (j) (1) ,D (j + 1)cj R logj R +
j!
2j R
uniformly on j,
with some c = c(D) > 0. At last, the power series expansion f (j) (s) =
gives (5) in the range |s − 1| 1/ log R.
3.3
P∞
u=0
f (j+u) (1)
(s
u!
− 1)u
Theorems 2 and 3
The following algebraic lemma (see Bump [2], Section 2.2) is necessary for the proofs.
0
Lemma (E. D. Littlewood). Let {αi }di=1 and {βj }dj=1 be non-zero complex numbers, and
Q 0
Q
define A = di=1 αi , B = dj=1 βj . Then
d
d Y
Y
(1 − αi βj x)−1= (1 − ABxd )−1
i=1 j=1
∞
X
Sk1 ,...,kd−1 (α)Sk1 ,...,kd−1 (β)xk1 +2k2 +3k3 +···+(d−1)kd−1
k1 ,...,kd−1 =0
if d = d0 , and
0
d Y
d
Y
(1 − αi βj x)−1=
i=1 j=1
if d <
∞
X
Sk1 ,...,kd−1 (α)Sk1 ,...,kd ,1,...,1 (β)Akd xk1 +2k2 +3k3 +···+dkd
k1 ,...,kd =0
d0 ,
with
d−1+k +···+k
1
α
d−1
1
d−2+kd−1 +···+k2
α
1
..
Sk1 ,...,kd−1 (α) = .
1+kd−1
α
1
1
...
...
..
.
...
...
d−1+k
+···+k
1
d−1
αd
α1d−1 . . .
d−2+kd−1 +···+k2 d−2
...
αd
α1
.
..
..
/ ..
.
.
α
1+k
...
1
αd d−1
1
...
1
16
αdd−1 αdd−2 .. .
. αd 1 The polynomials Sk1 ,...,kd−1 (α) are called Schur’s polynomials.
Let A(n) and B(n) be the totally multiplicative functions defined by A(p) =
QdB
B(p) = j=1
βj (p). By Littlewood’s lemma we have
Y
1−
p
p
1−
i=1 αi (p)
and
∞
|A(p)|2 |A(p)|2 X
An Y
=
1
−
L(s, A ⊗ Ā)
pdA s n=1 ns
pdA s
p
∞
X
=
Y
QdA
|a(n1 , . . . , ndA −1 )|2
dA −1 s
2
n1 ,··· ,ndA −1 =1 (n1 n2 · · · ndA −1 )
, (35)
∞
|B(p)|2 Bn Y
|B(p)|2 X
=
1
−
L(s, B ⊗ B̄)
pdB s n=1 ns
pdB s
p
=
and
Y
1−
p
∞
X
|b(n1 , . . . , ndB −1 )|2
dB −1 s
2
n1 ,··· ,ndB −1 =1 (n1 n2 · · · ndB −1 )
∞
X
a(n1 , . . . , nd−1 )b(n1 , . . . , nd−1 )
s
(n1 n22 · · · nd−1
d−1 )
1 ,··· ,nd−1 =1
A(p)B(p) L(s, A ⊗ B) =
pds
n
, (36)
(37)
if dA = dB = d; we have
L(s, A ⊗ B) =
∞
X
a(n1 , . . . , ndA −1 )b(n1 , . . . , ndA , 1, · · · , 1)A(ndA )
(n1 n22 · · · nddA
)s
A
n1 ,··· ,ndA =1
(38)
if dA < dB . Where the coefficients a(·) and b(·) are multiplicative in every entry and are defined
as
a(pk1 , . . ., pkd−1 ) = Sk1 ,...,kd−1 (α(p)) , b(pk1 , . . ., pkd−1 ) = Sk1 ,...,kd−1 (β(p)).
Theorem 2. In this case we assume that L(s, A ⊗ Ā) and L(s, B ⊗ B̄) satisfy (R), hence we
can prove
X Bn
X An
,DA (RA x) ,
,DB (RB x)
n
n
n≤x
n≤x
by (30) and using the Iwaniec’s idea. We can assume dA ≤ dB . By (35), (36) and by the
assumptions of Theorem 2 giving |A(n)|, |B(n)| ≤ 1 for every n, we have
X
|a(n1 , . . . , ndA −1 )|2
−1
n1 n22 · · · nddA
d −1
A −1
n1 n22 ···ndA −1 ≤x
P
≤
X
d|n Ad
n
n≤x
log x
X An
n≤x
n
,DA (RA x)
(39)
A
and
|b(n1 , . . . , ndA , 1, . . . , 1)|2
X
d
n1 n22 ···ndA ≤x
n1 n22 · · · nddA
A
X
≤
d −1
|b(n1 , . . . , ndB −1 )|2
n1 n22 ···ndB −1 ≤x
A
n1 n22 · · · nddBB −1
−1
B
P
≤
X
d|n Bd
n
n≤x
17
log x
X Bn
n≤x
n
,DB (RB x) .
(40)
Suppose now that dA = dB = d. By (37), (39) and (40) it follows that
logj n1 n22 · · · nd−1
d−1
X
a(n1 , . . . , nd−1 )b(n1 , . . . , nd−1 ) d−1
2
n1 n2 · · · nd−1
n1 n22 ···nd−1
≤x
d−1
≤ logj x
X
|a(n1 , . . . , nd−1 )|2 1/2 d−1
n1 n22 · · · nd−1
n1 n22 ···nd−1
≤x
d−1
X
|b(n1 , . . . , nd−1 )|2 1/2
n1 n22 · · · nd−1
d−1
n1 n22 ···nd−1
≤x
d−1
,DA ,DB (RA RB x) logj x
(41)
uniformly on j. We define
L∗ (s, A ⊗ B) =
Y
1−
p
A(p)B(p) L(s, A ⊗ B).
pds
1
The function L(s, A ⊗ B) has a polynomial behavior on the strip 1 − 2d
< σ < 2, and the same
Q
ds
holds for
1 − A(p)B(p)/p
since |A(p)|, |B(p)| ≤ 1 and d > 1 by hypothesis. Therefore,
also L∗ (s, A ⊗ B) has a polynomial behavior in that strip. We define
I
(j)
1
(x) =
2πi
Z
2+i∞
2−i∞
(j)
smA⊗B L∗ (1 + s, A ⊗ B)
s(s + 1) · · · (s + r)
xs
ds
with r = r(DA , DB ) independent of j and sufficiently large to assure the convergence of the
integral. As for Theorem 1, from (37) and (41) we get
I (j) (x) ,DA ,DB (j + 1)cj RA RB x logj x
(42)
uniformly on j. Moving the integration line to σ = −1/2d we have
(j)
I
(j)
smA⊗B L∗ (1 + s, A ⊗ B)
(x) =
r!
|s=0
+O
j!(R R )c1 A B
2j x1/2d
,
(43)
for some positive constant c1 = c1 (DA , DB ). Choosing x = (RA RB )2d(c1 +1) , the claim follows
by (42), (43) and some easy algebraic manipulations.
Suppose now that dA < dB . Then, by (38) and (40), we get
logj n1 n22 · · · nddA
A
X
a(n1 , . . . , ndA −1 )b(n1 , . . . , ndA , 1, · · · , 1)A(ndA ) d
n1 n22 · · · ndA
A
d
n1 n22 ···ndA ≤x
A
≤ logj x
|a(n1 , . . . , ndA −1 )|2 1/2 X
d
n1 n22 ···ndA ≤x
n1 n22 · · · nddA
A
|b(n1 , . . . , ndA , 1, · · · , 1)|2 1/2
X
d
n1 n22 ···ndA ≤x
n1 n22 · · · nddA
A
A
A
,DA ,DB (RA RB x) logj x
uniformly on j. The claim of Theorem 2 again follows from this estimate.
Theorem 3. Following the notations of the axioms, let L(s, A) =
SS (x) =
X
n≤x
(n,S)=1
18
|an |2
,
n
P∞
n=1 an n
−s
and consider
as we done for the proof of Theorem 1. By Proposition 6 we have |an |2 ≤ An , so that
c
SS (x) ,DA RA x
holds from (30) for every > 0, for some c = c(D) > 0. Similarly to (32), in this case we obtain
X∗ R2 (d) SS2 (x) log3 x
d≤x2
(d,S)=1
d
SS (x2 ),
where ∗ is convergent by (R’); from this recursive upper bound and with a suitable choice of
S we get
X |an |2
,DA (RA x) ;
(44)
n
n≤x
P
therefore, under the hypotheses of Theorem 3, estimates (39) and (40) are easy consequences of
(44), and upper bounds (9) follows from these ones as in Theorem 2.
A c k n o w l e d g m e n t s: This paper is a part of my Ph.D. thesis; I would like to thank
Prof. A. Perelli, my Ph.D. advisor. Moreover, I thank the referee for his careful reading and for
his suggestions which have considerably improved the previous version of this paper.
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