Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

0.32 MB·English

by  Étienne Fouvry

GAUSSIAN DISTRIBUTION FOR THE DIVISOR FUNCTION AND HECKE
EIGENVALUES IN ARITHMETIC PROGRESSIONS
E´TIENNEFOUVRY,SATADALGANGULY,EMMANUELKOWALSKI,ANDPHILIPPEMICHEL
Abstract. We show that, ina restricted range, the divisor function of integers in residue classes modulo
aprimefollowsaGaussiandistribution, andasimilarresultforHecke eigenvalues of classicalholomorphic
3
cuspforms. Furthermore,weobtainthejointdistributionofthesearithmeticfunctionsintworelatedresidue
1
classes. Theseresultsfollowfromasymptoticevaluationsoftherelevantmoments,anddependcruciallyon
0
resultsontheindependence ofmonodromygroupsrelatedtoproductsofKloostermansums.
2

n
a
J
2 1. Introduction

The distribution of arithmetic functions in arithmetic progressions is one of the cornerstones of modern
]
T analyticnumbertheory,withaparticularfocusonissuessurroundinguniformitywithrespecttothemodulus
N (see [7] for a recent survey). Besides the case of primes in arithmetic progressions, much interest has been
. devoted to the divisor function d(n) and higher-divisor functions, in particular because – in some precise
h
sense – a goodunderstanding of a few of these is equivalent to knowledgeabout the primes themselves (see,
t
a e.g., [6, Th´eor`eme4]).
m
The consideration of the second moment for primes p X in arithmetic progressions to moduli q
≤ ≤
[ Q X/(logX)A leads to the Barban-Davenport-Halberstam theorem (see, e.g., [13, Th. 17.2]), which has
≤
been refined to an asymptotic formula for Q=X by Montgomery [20]. Similarly, Motohashi [21] evaluated
1
v asymptoticallythe varianceofthedivisorfunctiond(n)forn X inarithmeticprogressionsmoduloq X.
≤ ≤
4 We will show that one can determine an asymptotic distribution for the divisor function d(n) for n X
1 in arithmetic progressions modulo a single prime p, provided however that X is a bit smaller than p2. ≤
2
0 Theorem 1.1 (Central Limit Theorem for the divisor function). Let w be a non-zero real-valued smooth
1. function on R with compact support in ]0,+ [ and with L2 norm w . For a prime p, let
∞ k k
0 n
3 Sd(X,p,a)= d(n)w ,
X
1 nX≥1 (cid:16) (cid:17)
: n≡amodp
v
i and
X
1 n 1 +∞ x
r M (X,p)= d(n)w (logx+2γ 2logp)w dx (1.1)
a d p X − p2 − X
nX≥1 (cid:16) (cid:17) Z0 (cid:16) (cid:17)
1 n 1
= d(n)w +O X(logX) ,
p X p2
nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
where γ is the Euler constant. For a F×, let
∈ p
S (X,p,a) M (X,p)
d d
E (X,p,a)= − .
d (X/p)1/2
Date:December 11,2013.
2010 Mathematics Subject Classification. 11F11,11F30, 11T23,11L05,60F05.
Keywords and phrases. Divisorfunction,Heckeeigenvalues,Fouriercoefficientsofmodularforms,arithmeticprogressions,
centrallimittheorem,Kloostermansums,monodromygroup,Sato-Tate equidistribution.
Ph. M.waspartiallysupportedbytheSNF(grant200021-137488) andtheERC(AdvancedResearchGrant228304);E´.F.
thanks ETHZu¨rich,EPFLausanneandtheInstitut UniversitairedeFranceforfinancial support;S.G.thanks EPFLausanne
forfinancialsupport.
1
Let Φ(x) 1 be any real-valued function, such that
≥
Φ(x) + as x + , Φ(x)=O (xǫ),
ǫ
−→ ∞ → ∞
for any ǫ > 0 and x 1. For any prime p, let X = p2/Φ(p). Then as p + over prime values, the
≥ → ∞
random variables
E (X,p,a)
d
a
7→ w 2π−2(logΦ(p))3
k k
on F×, with the uniform probability on F×, converge in distribution to a standard Gaussian with mean 0 and
p p p
variance 1, i.e., for any real numbers α<β, we have
1 a F× α Ed(X,p,a) β 1 βe−t2/2dt.
p−1(cid:12)n ∈ p | ≤ kwk 2π−2(logΦ(p))3 ≤ o(cid:12) p−→→∞ √2π Zα
(cid:12) (cid:12)
In fact, our results(cid:12)are more general, inpthree directions: (1) w(cid:12)e will consider, in addition to the divisor
function, the Fourier coefficients of any classical primitive holomorphic modular form f of level 1 (e.g., the
Ramanujan τ function); (2) we will compute the moments of the corresponding random variables and, for
a fixed moment, obtain a meaningful asymptotic in a wider range of X and p; (3) we will also consider the
joint distribution of
a (E (X,p,a),E (X,p,γ(a)))
d d
7→
when γ is a fixed projective linear transformation (e.g., γ(a) = a+1, γ(a) = 2a, γ(a) = a, γ(a) = 1/a,
−
whichillustratevariousinterestingphenomena.) Foralltheseresults,thecrucialingredientsaretheVoronoi
summationformula,andthe RiemannHypothesis overfinite fields, in the formofresults ofindependence of
monodromy groups of sheaves related to Kloosterman sums.
We nowintroduce the notationto handle these moregeneralproblems. As inthe statementabove,we fix
a non-zero smooth function w : R R, with compact support in [w ,w ] with 0 < w < w < + . For
0 1 0 1
→ ∞
any modulus c 1, let
≥ n
S (X,c,a)= d(n)w .
d
X
nX≥1 (cid:16) (cid:17)
n≡amodc
This sum has, asymptotically, a natural main term (see, e.g., [17]) which we denote by M (X,c), and
d
which coincides with M (X,p) when c = p is prime (see (2.8) below). The number of terms in S (X,c,a)
d d
is X/c and the square root cancellation philosophy suggests that its difference with the main term should
≈
be of size
Sd(X,c,a) Md(X,c) (X/c)21Xǫ, (1.2)
− ≪
as long as X/c gets large. Thus the map
S (X,c,a) M (X,c)
Z : a (Z/cZ)× E (X,c,a)= d − d .
∈ 7→ d (X/c)1/2
is a natural normalized error term that we wish to study as a random variable on (Z/cZ)× equipped with
the uniformprobabilitymeasure(here andbelow,we sometimesomit the dependency onpandX to lighten
the notation Z).
Similarly, consider a primitive (Hecke eigenform) holomorphic cusp form f of even weight k and level 1
(these restrictions are mainly imposed for simplicity of exposition). We write
f(z)= ρ (n)n(k−1)/2e(nz)
f
n≥1
X
its Fourier expansion at infinity, so that ρ (1) = 1 and ρ (n) is the eigenvalue of the Hecke operator T(n)
f f
(suitably normalized). We let
n 1 n
S (X,c,a)= ρ (n)w , M (X,c)= ρ (n)w ,
f f f f
X c X
n≡aX(modc) (cid:16) (cid:17) nX≥1 (cid:16) (cid:17)
S (X,c,a) M (X,c)
f f
E (X,c,a)= − ,
f (X/c)1/2
2
for c 1 and any integer a. Note that, in this case, the integral representation
≥
1 1 2+i∞
M (X,c)= wˆ(s)XsL(s,f)ds
f
c × 2πi
Z2−i∞
in terms of the Mellin transform wˆ of w shows that the main term is very small, namely
M (X,c) c−1X−A (1.3)
f f,A
≪
for every positive A, uniformly for c 1 and X 1.
≥ ≥
We will study the distribution of
a E (X,p,a), a E (X,p,a)
f d
7→ 7→
for p prime using the method of moments. Thus, for any integer κ 1, we define
≥
1
(X,c;κ)= E (X,c,a)κ, ⋆=d or f. (1.4)
⋆ ⋆
M c
amodc
(aX,c)=1
The first moment is very easy to estimate, and besides Motohashi’s work (which considers the average
of (X,c;2) over c X), the second moment has recently been discussed by Blomer [2], Lu¨ [18] and
d
M ≤
Lau–Zhao [17]. In particular, Lau and Zhao obtained an asymptotic formula in the range X1/2 < c < X
(see (1.10) below; note that the range c<X1/2 seems to be much more delicate.)
We will evaluate any moment, in a suitable range. Precisely, in 3 we will prove:
§
Theorem 1.2. Let the notation be as above, with ⋆=d, the divisor function, or ⋆=f, f a Hecke form of
weight k and level 1. Let p be a prime number. Then, for every integer κ 1, for every positive δ, for every
≥
positive ǫ, for every X satisfying
2 X1/2 p<X1−δ, (1.5)
≤ ≤
we have the equality
p2 κ/2 X 1/2+ǫ
(X,p;κ)=C (κ)+O p−1/2+ǫ + , (1.6)
M⋆ ⋆ X p2
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
where the implied constant depends on (δ,ǫ,κ,f,w), and the constant C (κ) is given by
⋆
κ/2
C (κ)=c m , (1.7)
⋆ ⋆,w κ
with
0 if κ is odd,
m = (1.8)
κ
 κ! if κ is even,
2κ/2(κ/2)!
and 
(4π)k p2
c = w 2 f 2 , c =P log , (1.9)
f,w d,w w
k k k k Γ(k) X
(cid:16) (cid:17)
for some polynomial P (T) R[T], depending only on w, of degree 3 with leading term 2 w 2π−2T3. Here,
w
∈ k k
for a cusp form f, the L2-norm of f is computed with respect to the probability measure
3dxdy
π y2
on SL (Z) H, and the L2-norm of w is computed with respect to the Lebesgue measure on R.
2
\
Remark 1.3. In the case κ = 2, and in the range X1/2 c X, Lau and Zhao [17, Theorem 1 (2)] have
≤ ≤
obtained
1 c c1/2 2 c 1 X 1 ϕ(ℓ)
6 4
ρ (n) =c +O d(c)+ , (1.10)
c X1/2 f f X c2 ℓ
Xa=1(cid:12) n≡aX(modc) (cid:12) (cid:16)(cid:16) (cid:17) (cid:16) (cid:17) Xℓ|c (cid:17)
(cid:12) 1≤n≤X (cid:12)
(cid:12) (cid:12)
for any modulus c 1 (not only primes), and a similar result for the divisor function.
≥
3
We will make further comments on this result after the proof, in Section 3.5. Since m is the κ-th
κ
moment of a Gaussian random variable with mean 0 and variance 1, we obtain the following, which implies
Theorem 1.1 in the case ⋆=d:
Corollary 1.4 (Central limit theorem). Let Φ(x) 1 be any real-valued function, such that
≥
Φ(x) + as x + , Φ(x)=O (xǫ),
ǫ
−→ ∞ → ∞
for any ǫ>0, uniformly for x 1. For any prime p, let X =p2/Φ(p). Then as p + over prime values,
≥ → ∞
the random variables
E (X,p,a)
⋆
a
7→ √c⋆,w
on F× converge in distribution to a standard Gaussian with mean 0 and variance 1.
p
As far as we know, this is the first result of this type. We will prove this in Section 3, and give further
comments, in Section 3.6.
Amongthenaturalgeneralizationsofthisresult,weconsidernextthefollowingone: givenamapa γ(a)
7→
on F×, what is the asymptotic joint distribution of
p
a (E (X,p,a),E (X,p,γ(a))) ?
⋆ ⋆
7→
We study this when γ is given by a fractional linear transformation. Precisely, let
a b
γ = GL (Q) M (Z) (1.11)
c d ∈ 2 ∩ 2
(cid:18) (cid:19)
be a fixed invertible matrix with integral coefficients. For p ∤ detγ, the matrix γ has a canonical reduction
modulo p in PGL (F ), which we denote by π (γ). In the usual manner, γ (or π (γ)) defines a fractional
2 p p p
linear transformation on P1 by
F
p
az+b
z P1F γ z = .
∈ p 7→ · cz+d
By Corollary 1.4, we know that, in the range of validity of this result, both
E (X,p,a) E (X,p,γ a)
⋆ ⋆
Z : a and Z γ : a · , (1.12)
7→ √c⋆,w ◦ 7→ √c⋆,w
seen as random variables defined on the set
a F a,γ a=0,
p
{ ∈ | · 6 ∞}
convergeto the normallaw. We then wish to know the asymptotic jointdistribution ofthe vector(Z,Z γ),
◦
and we study this issue, as before, using moments.
For κ and λ positive integers, let
1
(X,p;κ,λ;γ):= E (X,p,a)κE (X,p,γ a)λ, (1.13)
⋆ ⋆ ⋆
M p ·
aX∈Fp
a, γ·a6=0,∞
be the mixed moment of order (κ,λ).
In analogy with Theorem 1.2, we will estimate these moments in 4. To state the result, we note that if
§
γ is diagonal, there is a unique triple of integers (α ,γ ,γ ), such that we have the canonical form
γ 1 2
γ 0
γ =α 1 , γ 1 and (γ ,γ )=1. (1.14)
γ 0 γ2 1 ≥ 1 2
(cid:18) (cid:19)
We further introduce the arithmetic functions
ρ (p)ρ (pα−1) d(p)d(pα−1)
ρ = ρ (pα) f f , ρ = d(pα) , (1.15)
a,f f − p+1 a,d − p+1
pYαka(cid:16) (cid:17) pYαka(cid:16) (cid:17)
for a 1, and ρ =0 for a<0, ρ =ρ for a<0. For ⋆=f, we also define the constant
≥ a,f a,d −a,d
c = f 2(4π)kΓ(k)−1. (1.16)
f
k k
Our result is:
4
Theorem 1.5. Let γ be defined by (1.11).
(1) For every integers κ and λ, for every δ and ǫ>0, for every prime p p (γ) and X satisfying (1.5),
0
≥
there exists C (κ,λ,γ) such that
⋆
p2 (κ+λ)/2 X 1/2+ǫ
M⋆(X,p;κ,λ;γ)=C⋆(κ,λ,γ)+O p−12+ǫ X + p2 . (1.17)
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
(2) If γ is non-diagonal, then
C (κ,λ,γ)=C (κ)C (λ). (1.18)
⋆ ⋆ ⋆
(3) If γ is diagonal, and written in the canonical form (1.14), then
0 if κ+λ is odd,
C⋆(κ,λ,γ)= ν! κ λ mκ−νmλ−ν(c⋆,w)κ+2λ−ν(c˜⋆,w,γ)ν, otherwise, (1.19)
0≤ν≤Xmin(κ,λ) (cid:18)ν(cid:19)(cid:18)ν(cid:19)
ν≡κ≡λmod2
where 
∞
c˜ =c ρ w(γ t)w(γ t)dt ,
f,w,γ f γ1γ2,f 1 2
(cid:16)Z−∞ (cid:17)
and for ⋆=d, we have
p2
c˜ =P log
d,w,γ γ1γ2,w X
(cid:16) (cid:17)
for some polynomial P (T) R[T], of degree 3 and with coefficient of T3 given by
γ1γ2,w ∈ ≤
2 ∞
ρ w(γ t)w(γ t)dt T3.
π2 γ1γ2,d 1 2
(cid:16)Z−∞ (cid:17)
In (1.17), the implied constant depends at most on (γ,δ,ε,κ,λ), and in (1.19), we make the convention
that 0ν =1 if ν =0.
Of course, if γ is the identity, we recover Theorem 1.2. More generally, we can now determine the joint
asymptotic distribution of (Z,Z γ) in the same range as Corollary 1.4.
◦
Corollary 1.6. Let Φ be a function as in Corollary 1.4, and let X =p2/Φ(p). Then, for ⋆ = f or d, as p
tends to infinity, the random vector (Z,Z γ) converges in distribution to a centered Gaussian vector with
◦
covariance matrix
1 0
, if γ is not diagonal. (1.20)
0 1
(cid:18) (cid:19)
1 G
⋆,γ,w , if γ is diagonal, (1.21)
G 1
⋆,γ,w
(cid:18) (cid:19)
where the covariance G is given by
⋆,γ,w
ρ
G = γ1γ2,⋆ w(γ t)w(γ t)dt.
⋆,γ,w w 2 R 1 2
k k Z
Recallthatapair(X,Y)ofrandomvariablesisaGaussian vector ifandonly,foreverycomplexnumbers
α and β, the random variable αX+βY has a Gaussian distribution (see, e.g., [14, pp. 121–124]). If (X,Y)
is a Gaussian vector, its covariance matrix cov(X,Y) is defined by
E(X2) E(X)2 E(XY) E(X)E(Y)
cov(X,Y)= − − , (1.22)
E(XY) E(X)E(Y) E(Y2) E(Y)2
(cid:18) − − (cid:19)
where E denotes the expectation of a random variable. Recall also that a Gaussian vector (X,Y) has
independentcomponentsifandonlyifE(XY)=E(X)E(Y), i.e.,ifthecovariancematrixisdiagonal(see[14,
Theorem 16.4] for instance). Thus from Corollary 1.6 (noting that ρ =0 for any integer a=0), we get a
a,d 6 6
criterion for asymptotic independence of (Z,Z γ):
◦
5
Corollary 1.7. We adopt the notations and hypotheses of Corollary 1.6. Then as p tends to , the random
∞
variables Z and Z γ tend to independent Gaussian random variables, if and only if one of the following
◦
conditions holds:
(1) If γ is not a diagonal matrix, i.e., a γ a is not a homothety,
7→ ·
(2) If γ is a diagonal matrix and
∞
w(γ t)w(γ t)dt=0,
1 2
Z−∞
(3) If ⋆ = f, γ is a diagonal matrix in the from (1.14), and there exists a prime p and α 1 such that
≥
pα γ γ and such that
2 1
k
(p+1)ρ (pα)=ρ (p)ρ (pα−1).
f f f
Remark 1.8. (1)Corollary1.7showsforinstancethat,forp ,therandomvariablesa E (p2/Φ(p),p,a)
⋆
→∞ 7→
anda E (p2/Φ(p),p,γ a)convergetoindependentGaussianvariables,ifγ isoneofthefollowingfunctions
⋆
7→ ·
γ a=a+1, γ a= a, γ a=1/a.
· · − ·
+∞
The case of γ a = 2a is more delicate, since it depends on the value of the integral w(t)w(2t)dt.
· 0
For instance, this integral is zero when one has the inequalities w < w < 2w < 2w , where as before
0 1 0 1
R
supp(w) [w ,w ]. The possible dependency here reflects the obvious fact that if n amodp and d n,
0 1
⊂ ≡ |
then 2n 2amodp and d 2n.
≡ |
(2) We do not know if any primitive Hecke form f of level 1 exists for which Condition (3) in this last
corollary holds for some pα! Certainly the “easiest” way it could apply would be if, for some p, we had
ρ (p)=0,but the existence of a primitive cuspformof level1 anda prime p with ρ (p)=0 seems doubtful
f f
(e.g.,aconjectureofMaedasuggeststhatthecharacteristicpolynomialsofthe HeckeoperatorsT(p)inlevel
1 are irreducible.) On the other hand, if we extend the result to forms of fixed level N 1, it is possible to
≥
have ρ (p)=0 for some p (e.g., for weight k =2 and f corresponding to an elliptic curve.)
f
1.1. Sketch of the proof. We will sketch the proof in the case of cusp forms, which is technically a bit
simpler, though we present the actual proofs in a unified manner. For Theorem 1.2, the crucial starting
point is the Voronoi summation formula, as in [2, 17], which expresses E (X,c,a) for any c 1 in terms
f
≥
of sums weighted by some smooth function of the Fourier coefficients ρ (n) twisted by Kloosterman sums
f
S(a,n;c). One then seesthat the main contributionto this sumcomes fromthe n ofsize roughlyY =c2/X
(see Proposition2.1).
Consideringtheκ-thmoment,weobtainthereforeanaverageoveramodpofaproductofκKloosterman
sums S(a,n ;p), where all variables n are of size approximately p2/X. The sum over a F×, when the
i i ∈ p
variables n are fixed, can be evaluated using deep results on the independence of Kloosterman sheaves
i
(see Proposition 3.2). This allows us to gain a factor p1/2 compared with a direct application of the Weil
bound for Kloostermansums, except for special, well-understood, configurations of the n modulo p. These
i
configurationslead, by combinatorialarguments,to the Gaussianmain termof Theorem1.2. (Note that we
can take no advantage of the summation over the variables n , which turn out to have a short range in the
i
cases where our result is non-trivial, see Section 3.5.)
The study of mixed moments (see Theorem 1.5) has a lot of similarities. The only significant difference
lies in the study of the independence of Klostermansheaves,when some of them are twisted by the rational
transformation γ. However, Proposition 3.2 is general enough to show that these sheaves are dependent if
andonlyifweareinthe“obvious”cases. Themaintermsthenrequiresomecomputationsofintegralsusing
properties of the Bessel transforms.
1.2. Possible extensions. A Gaussian law similarly appears if one studies the random variable a
7→
E (X,p,P(a)), where P is a non–constant fixed polynomial with integer coefficients. The fact that P is
⋆
not necessarilya bijection on F does not affect the Gaussianbehavior. The proofof this extensionrequires
p
a suitable generalizationof Proposition 3.2.
It also seems that the present method can be extended to the study of the distribution of sums of the
shape
n
a S (X,p,K )= τ (n)K (n)w
⋆ a ⋆ a
7→ X
nX≥1 (cid:16) (cid:17)
6
where τ () is either d() or ρ (), and K (n) = K(an) for a fairly general trace function K as in [8]. The
⋆ f a
· · ·
shape of the analogue of Theorem1.2 would then depend on the nature of the geometric monodromy group
of a suitable “Bessel transform” of the sheaves underlying K().
·
Another natural extension, which we are currently considering, is that of coefficients of cusp forms on
higher-rank groups, and of higher divisor functions.
1.3. Notations. We use synonymously the notation f(x) g(x) for x X and f = O(g) for x X. We
≪ ∈ ∈
denote e(z)=e2iπz for z C. For c 1 and a, b integers,or congruence classes modulo c, the Kloosterman
∈ ≥
sum S(a,b;c) is defined by
ax+bx¯
S(a,b;c)= e
c
x(xXm,co)=d1c (cid:16) (cid:17)
where x¯ is the inverse of x modulo c. The normalized Kloosterman sum is defined by
S(a,b;c)
Kl (a,b;c)= ,
2 c1/2
and for (a,b,c)=1 it satisfies the Weil bound
Kl (a,b;c) d(c). (1.23)
2
| |≤
To lighten notations, we define
Kl (a;c):=Kl (a,1;c),
2 2
and recall the equality Kl (a,b;c)=Kl (ab;c), whenever (b,c)=1.
2 2
We will use the Bessel functions J , where k 2 is an integer, Y and K ; precise definitions can be
k−1 0 0
≥
found for instance in [11, App. B.4] and in [24].
2. Preliminaries
We gather in this section some facts we will need during the proof of the main results. The reader may
wish to skip to Section 3 and refer to the results when they are needed.
We begin with the Voronoi formula in the form we need:
Proposition 2.1 (Voronoi summation). Let ⋆ = f, for a cusp form f of level 1 and weight k, or ⋆ = d.
Let c be any positive integer, with c prime if ⋆=d. Then for any X 1 and for any integer a, we have the
≥
equality
X1/2 c 1/2 nX
E (X,c,a)= τ (n)W Kl (a,n;c ), (2.1)
⋆ c c ⋆ ⋆ c2 2 1
Xc1|c(cid:16) 1(cid:17) nX6=0 (cid:16) 1 (cid:17)
c1>1
where n runs on the right over non-zero integers in Z and
ρ (n) if n 1,
f
τ (n)= ≥ (2.2)
f
(0 otherwise,
τ (n)=d(n), (2.3)
d
| |
and
∞
W (y)=2πik w(u)J (4π√uy)du for y >0, (2.4)
f k−1
Z0
W (y)=0, for y <0,
f
∞
W (y)= 2π w(u)Y (4π√uy)du, for y >0, (2.5)
d 0
−
Z0
∞
W (y)=4 w(u)K (4π uy )du, for y <0. (2.6)
d 0
| |
Z0
p
In particular, if c=p, a prime, we have
X 1/2 nX
E (X,p,a)= τ (n)W Kl (a,n;p). (2.7)
⋆ p2 ⋆ ⋆ p2 2
(cid:16) (cid:17) nX6=0 (cid:16) (cid:17)
7
For the proof we recall the standard Voronoi summation formula (see, e.g., [13, p. 83] for ⋆=f and [13,
(4.49)] for ⋆=d, which we rewrite as a single sum over positive and negative integers instead of two sums).
Lemma 2.2. Let c be a positive integer and a an integer coprime to c.
(1) For any smooth function w compactly supported on ]0, [, we have
∞
an 1 n na
ρ (n)w(n)e = ρ (n)W e ,
f c c f f c2 − c
nX≥1 (cid:16) (cid:17) nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
if f is a cusp form of level 1 and weight k.
(2) For any smooth function w compactly supported on ]0, [, we have
∞
an 1 +∞ 1 n an
d(n)w(n)e = (logx+2γ 2logc)w(x)dx+ d(n)W e . (2.8)
c c − c | | d c2 − c
nX≥1 (cid:16) (cid:17) Z0 nX6=0 (cid:16) (cid:17) (cid:16) (cid:17)
Proof of Proposition 2.1. We consider the case of ⋆ = f, the divisor function being handled similarly (it is
easier since c is prime; the definition (1.1) of the main term is designed to cancel out the first main term
in(2.8)). Using orthogonalityofadditive characters,andseparatingthe contributionofthe trivialcharacter
from the others, we write
c−1
1 ab n bn
S (X,c,a)= e ρ (n)w e
f f
c − c X c
Xb=0 (cid:16) (cid:17)nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
1 n 1 ab n bn
= ρ (n)w + e ρ (n)w e ,
f f
c X c − c X c
nX≥1 (cid:16) (cid:17) 1≤Xb≤c−1 (cid:16) (cid:17)nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
which yields the expression
1 ab n bn
E (X,c,a)= e ρ (n)w e .
f (cX)1/2 − c f X c
1≤Xb≤c−1 (cid:16) (cid:17)nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
We split the second according to the value of the g.c.d d=(b,c), writing
d=(b,c),b=db ,c=dc ,
1 1
and note that
1<c c, 1 b <c .
1 1 1
≤ ≤
We then get
1 ab n bn
E (X,c,a)= e ρ (n)w e
f (cX)1/2 − c f X c
Xd|c 1≤Xb<c (cid:16) (cid:17)nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
(b,c)=d
1 ab n b n
1 1
= e ρ (n)w e .
(cX)1/2 − c f X c
1 1
Xc1|c 1≤Xb1<c1 (cid:16) (cid:17)nX≥1 (cid:16) (cid:17) (cid:16) (cid:17)
c1>1(b1,c1)=1
We can now apply Lemma 2.2 since (b ,c )=1, and we get
1 1
n b n X nX b n
1 1
ρ (n)w e = ρ (n)W e .
f X c c f c2 − c
nX≥1 (cid:16) (cid:17) (cid:16) 1 (cid:17) 1 nX≥1 (cid:16) 1 (cid:17) (cid:16) 1 (cid:17)
The proposition now follows since the terms with n<0 are identically zero for this case. (cid:3)
We will need some basic information on the behavior of the Bessel transforms W (y).
⋆
Proposition 2.3. Let w be a smooth function with support included in ]0,+ [. Let W (y) be one of the
⋆
∞
Bessel transforms of w as defined in Proposition 2.1, for some integer k 2 in the case ⋆=f of weight k.
≥
(1) The function W is smooth on R×, and for every A 0 and j 0, we have
⋆
≥ ≥
yjW(j)(y) min 1+ log y , y −A , (2.9)
⋆ A,j
≪ | | | |
for y =0. (cid:0) (cid:12) (cid:12) (cid:1)
6 (cid:12) (cid:12)
8
(2) We have
W = w , (2.10)
⋆
k k k k
where the L2-norm of W and w are computed in L2(R×) with respect to Lebesgue measure.
⋆
(3) More generally, for any two non-zero real numbers m and n, we have
∞ ∞
W (mt)W (nt)dt= w(mt)w(nt)dt.
⋆ ⋆
Z−∞ Z−∞
Proof. (1) (Compare, e.g., with [2, p. 280], [17, Lemma 3.1]) We begin with the case j =0. For y small, we
use the bounds
J (x) 1, Y (x) 1+ logx , K (x) 1+ logx
k−1 k 0 0
≪ ≪ | | ≪ | |
for 0<x 1 which immediately imply
≤
W (y) 1+ log y (2.11)
⋆
≪ | |
in all cases. (cid:12) (cid:12)
(cid:12) (cid:12)
To deal with the case where y 1, we first make the change of variable
| |≥
v =4π uy
| |
in the integrals (2.4) (resp. (2.5), (2.6)), so that we alwpays get
1 ∞ v2
W (y)= w vB (v)dv,
⋆ y 16π2y2 0
| |Z0 (cid:16) (cid:17)
where B =cJ , 0, cY or cK , for some fixed multiplicative constant c C.
0 k−1 0 0
∈
Wedenoteα=(16π2y2)−1. ToexploitconvenientlytheoscillationsoftheBesselfunctionsB weintegrate
0
by parts, using the relations (see [10, 8.472.3, 8.486.14])
(xν+1Z (x))′ =ǫxν+1Z (x), (2.12)
ν+1 ν
where
+1 if Z =J or Y ,
ν ν ν
ǫ=

 1 if Zν =Kν.
−
For ⋆=f, remembering that w vanishes at 0 and , we obtain, for instance, the equality
 ∞
c ∞
W (y)= 2αv2w′(αv2)+(1 k)w(αv2) J (v)dv (y >0).
f k
− y −
| |Z0 (cid:16) (cid:17)
By iterating ℓ 1 times, and then arguing similarly for ⋆ = d, we see that there exist coefficients ξ
ℓ,ν
≥
such that
1 ∞ ℓ
W (y)= ξ (αv2)νw(ν)(αv2) v−ℓ+1B (v)dv, (2.13)
⋆ ℓ,ν ℓ
y
| |Z0 (cid:16)νX=0 (cid:17)
where B =J , 0, Y or K corresponding to the different cases ⋆=f or ⋆=d, y >0 or y <0.
ℓ k−1+ℓ ℓ ℓ
Since w has compact support in [w ,w ], the above integral can be restricted to the interval
0 1
:= (w /α)1/2,(w /α)1/2 ,
0 1
I
and using the estimates1 (cid:2) (cid:3)
J (v) v−1/2, Y (v) v−1/2, K (v) v−1/2
k−1+ℓ ℓ ℓ ℓ ℓ ℓ
≪ ≪ ≪
for v 1, we obtain the inequality
≥
W⋆(y) y −1 v−ℓ+12dv y −1−ℓ/2+3/2 (2.14)
≪| | ≪| |
ZI
for y 1. Since ℓ 0 is arbitrary,this gives the result for j =0.
| |≥ ≥
We can reduce the general case to j =0 using the formulas (see [10, 8.472.2, 8.486.13])
xZ′(x)=νZ (x) xZ (x),
ν ν − ν+1
1Forthelastone,oneknowsinfactthatKℓ(v)decays exponentiallyfastforv→+∞.
9
from which it follows that
d ∞ ν ∞ ∞
y w(u)Z (4π√uy)du = w(u)Z (4π√uy)du 2π√y w(u)√uZ (4π√uy)du.
ν ν ν+1
dy 2 −
(cid:16)Z0 (cid:17) Z0 Z0
Applying the previous method to the relevant Bessel functions then leads to
yW′(y) min(1+ log y ,y−A)
⋆ ≪⋆,A | | ||
and by induction a similar argument deals with higher derivatives.
(2) In the case ⋆=f, the identity
+∞ +∞
W (u)2du= w(u)2du= w 2
f
k k
Z0 Z0
is a direct consequence of the unitarity of the Hankel transform, i.e., of the Fourier transform for radial
functions (see, e.g., [17, Lemma 3.4]). The case ⋆ = d is less classical, although it is formally similar, the
hyperbolas xy = r replacing the circles x2 +y2 = r2 (see [13, 4.5]). We use a representation-theoretic
§
argument to get a quick proof. The unitary principal series representation ρ = π(0) of PGL (R) (in the
2
notationof[4,p. 10])canbedefinedbyitsKirillovmodelwithrespecttotheadditivecharacterψ(x)=e(x),
which is a unitary representation of PGL (R) on L2(R×, x−1dx). In this model, the unitary operator
2
| |
0 1
T =ρ −
1 0
(cid:16)(cid:18) (cid:19)(cid:17)
on L2(R×, x−1dx) is given by
| |
dt
Tϕ(x)= ϕ(t) (xt) ,
R× J t
Z | |
where is the so-called “Bessel function” of ρ (with respect to ψ, see [4, Th. 4.1]). By [4, Prop. 6.1, (ii)]
J
(see also [1, 6, 21]), we have
§ §
2π√uY (4π√u) for u>0,
0
(u)= −
J (4 uK0(4π u) for u<0.
| | | |
Hence by (2.5) and (2.6), we see that p p
√xw(x) if x>0
W (y)= y −1/2T(ϕ)(y), where ϕ(x)= (2.15)
d
| | (0 if x<0.
The unitarity of T means that
dy dx
T(ϕ)(y)2 = ϕ(x)2 ,
R×| | y R×| | x
Z | | Z | |
i.e.,
+∞
W (y)2dy = w(x)2dx= w 2.
d
ZR×| | Z0 | | k k
(3)Weconsiderdifferentcases. Ifmn>0,changingtto tallowsustoassumethatmandnarepositive.
−
Then a simple polarization argument from (2.10) shows that
+∞ ∞
W (mt)W (nt)dt= w (u)w (u)du, (2.16)
⋆ ⋆ m n
Z−∞ Z−∞
where u w (u) is the function for which the Bessel transform of is t W (mt) and similarly for w (u).
m ⋆ n
But it is7→immediate that w (u)=(1/m)w(u/m), and therefore (2.16) g7→ives the result.
m
If mn<0, then since the support of w is contained in [0,+ [, we have w(mt)w(nt)=0 for all t, hence
∞
w(mt)w(nt)dt =0,
R
Z
and we must show that the integral of W (mt)W (nt) is also zero. If ⋆=f, a cusp form, this is immediate
⋆ ⋆
since W (y)=0 for y <0, so that W (mt)W (nt)=0 for all t.
f f f
10