On the functional equation of the standard twist associated with half-integral weight cusp forms

0.21 MB·

by  J. Kaczorowski

ON THE FUNCTIONAL EQUATION OF THE STANDARD TWIST
ASSOCIATED WITH HALF-INTEGRAL WEIGHT CUSP FORMS
J.KACZOROWSKI and A.PERELLI
7
Abstract. Let F(s) be the normalized Hecke L-function associated with a cusp form of
1
0 half-integral weight κ and level N. We show that the standard twist F(s,α) of F(s) satisfies a
2 functional equation reflecting s to 1 s. The shape of the functional equation is not far from
−
n a standard Riemann type functional equation of degree 2; actually, it may be regarded as a
a
degree 2 analog of the Hurwitz-Lerch functional equation. We also deduce some result on the
J
order of growth on vertical strips and on the distribution of zeros of F(s,α).
4
1
Mathematics Subject Classification (2000): 11M41, 11F66, 11F37

] Keywords: Selberg class, half-integral weight modular forms, standard twist, functional
T
equations.
N
.
h 1. Introduction
t
a
m
1.1. Set up of the problem. In [8], [10] and [12] we introduced and studied the standard
[ twist

1 ∞ a(n)
v F(s,α) = e( αn1/d) (1.1)
ns −
9
n=1
2 X
of any function F(s) from the extended Selberg class; here a(n) and d are, respectively, the
9
3 Dirichlet coefficients and the degree of F(s), e(x) = e2πix and α > 0 (see below for definitions).
0 In particular, it turns out that F(s,α) has meromorphic continuation to C and polynomial
.
1
growth on vertical strips, and some basic information is known about its polar structure. The
0
7 standard twist plays a relevant role inside the Selberg class theory and, moreover, has interest-
1 ing applications to the classical L-functions. For example, it gives asymptotic expansions for
:
v nonlinear exponential sums associated with such L-functions and provides a theoretical expla-
i
X nation of the Ω-results for their coefficients; see [10]. We give a more detailed description of the
r properties of F(s,α) below, in the framework of the L-functions associated with half-integral
a
weight cusp forms.
The extended Selberg class ♯ consists of non identically vanishing Dirichlet series F(s),
S
absolutely convergent for σ > 1, such that (s 1)mF(s) is entire of finite order for some integer
−
m and satisfying a functional equation of type
r r
Qs Γ(λ s+µ )F(s) = ωQ1 s Γ(λ (1 s)+µ )F(1 s) (1.2)
j j − j j
− −
j=1 j=1
Y Y
with ω = 1, Q > 0, λ > 0 and µ 0. We refer to Selberg [21], Conrey-Ghosh [3], to our
j j
| | ℜ ≥
survey papers [9], [6], [18], [19], [20] and to the forthcoming book [14] for definitions, examples
and the basic theory of the class ♯. Here we recall that degree d and conductor q of F(s) are
S
defined, respectively, by
r r
d = 2 λ q = (2π)dQ2 λ2λj; (1.3)
j j
j=1 j=1
X Y
further definitions will be introduced later on.
1
2
Let f be a cusp form of half-integral weight κ = k/2 and level N, where k > 0 is an odd
integer and 4 N, and L (s) be the associated Hecke L-function. Then L (s) is entire and
f f
|
satisfies the functional equation
√N
s
Λf(s) = ωΛf∗(κ s), where Λf(s) = Γ(s)Lf(s), (1.4)
− 2π
ω = 1 and f∗ is related to f by the slash operator. Note t(cid:0)hat L(cid:1)f∗(s) is also entire and has
| |
properties similar to L (s). We refer to Ogg [16] and to Section 4.3 of Miyake [15] for the
f
basic analytic theory of modular forms, and to Bruinier [1] for a detailed exposition of the
half-integral weight case.
Comparing functional equations (1.2) and (1.4), it is clear that the function L (s) does
f
not belong to the extended Selberg class ♯. However, it come close after the normalization
S
s s+ κ 1. Indeed, writing
−
7→ 2
κ 1 κ 1
F(s) = Lf(s+ − ) and F∗(s) = Lf∗(s+ − ), (1.5)
2 2
functional equation (1.4) becomes
√N κ 1 √N κ 1
s 1 s
Γ(s+ − )F(s) = ω − Γ(1 s+ − )F∗(1 s). (1.6)
2π 2 2π − 2 −
Although this(cid:0)is not(cid:1) exactly of the form (1.(cid:0)2) (F(cid:1)(s) is not necessarily equal to F(s), and for
∗
k = 1 we have κ 1 = k 2 < 0), most results in the Selberg class theory hold in this case as
− −
2 4
well. Note that the other requirements of ♯ are satisfied by F(s), as it can be checked by the
S
argument in the proof of Theorem 5.1 and Corollary 5.2 of Iwaniec [4] (see also the proof of the
Theorem in Kaczorowski et al. [7] and on p.217-218 of Carletti et al. [2]). In particular, both
F(s) and F (s) are absolutely convergent for σ > 1. Note also that, in view of (1.6), F(s) has
∗
degree d = 2 and conductor q = N.
From now on we consider cusp forms f of weight κ and level N as above, and the normalized
Hecke L-functions F(s) and F (s) as in (1.5). We denote by a(n) and a (n) their Dirichlet
∗ ∗
coefficients, respectively, and let F(s,α) be the standard twist of F(s), defined by (1.1) with
d = 2 in this case. Then, minor formal modifications to the proof of Theorem 1 of [10], or of
Theorems 1 and 2 of [12], give the following properties of F(s,α). Writing
n = Nα2/4 and Spec(F) = α > 0 : a (n ) = 0 , (1.7)
α ∗ α
{ 6 }
where a (n ) = 0 if n N, we have that F(s,α) is entire if α Spec(F). If α Spec(F),
∗ α α
then F(s,α) is meromor6∈phic over C with at most simple poles at6∈ ∈
3 ℓ
s = ℓ = 0,1,... (1.8)
ℓ
4 − 2
(since the internal shift θ vanishes in this case, see its definition in [12]) and
F
a (n )
∗ α
res F(s,α) = c (F) c (F) = 0. (1.9)
s=s0 0 n1/4 0 6
α
Moreover, in all cases F(s,α) has polynomial growth on vertical strips, although the bounds
obtained in [10] and [12] are poor from a quantitative point of view.
The main result of this paper is that F(s,α) satisfies a functional equation. Before stating
the main theorem, we recall that in [11] we proved that if a general F ♯ has degree 2
∈ S ≥
and satisfies the Ramanujan conjecture (i.e. a(n) nε), then F(s,α) does not belong to ♯.
≪ S
Actually, a variant of the arguments in [11], which we sketch in the Appendix below, proves
under the same assumptions the stronger assertion that F(s,α) does not satisfy a functional
equation of type (1.2). Nevertheless, in the case of half-integral weight modular forms we prove
3
that the standard twist F(s,α) satisfies a functional equation, not exactly of Riemann type but
still reflecting s into 1 s.
−
Remark 1. The Ramanujan conjecture is crucial for our results in [11]. Indeed, for certain
Dirichlet L-functions, say L (s), the standard twists L (s,α) are still degree 1 functions of ♯
1 1
S
for suitable values of α. But such L (s) can be lifted to degree 2 functions L (s) by letting
1 2
s 2s 1/2, and for the same values of α their standard twists L (s,α) satisfy a degree 2
2
7→ −
functional equation of type (1.6), with κ = 1/2 or κ = 3/2. Moreover, when κ = 3/2 bothL (s)
2
and L (s,α) belong to ♯. However, such L (s)’s do not satisfy the Ramanujanconjecture, thus
2 2
S
showing that this hypothesis is crucial in [11]; also, it turns out that they are related with half-
integral weight modular forms. We refer to Section 2 of [13] for the basic properties of lifts
in ♯. Apart from these special cases, we don’t know examples of functions F(s) satisfying a
S
functional equation of type (1.2) of degree d 2 with standard twist F(s,α) also satisfying a
≥ (cid:3)
functional equation of the same type.
1.2. Functional equation. Let f be a cusp form of half-integral weight κ = k/2 and level
N, F(s) and F (s) be as in (1.5), F(s,α) be the standard twist of F(s), and a (n) be the
∗ ∗
coefficients of F (s). We write
∗
k = 2h+1 with h 0,1,2... , µ = (2h 1)/4, h = max(0,h 1), (1.10)
∗
∈ { } − −
ν = √n with n = 1,2,... and ν = √n with n as in (1.7). (1.11)
α α α
±
Moreover, for ℓ = 0,...,h we also write
∗
F (s,α) = e iπsF+(s,α)+eiπsF (s,α), (1.12)
ℓ∗ − ℓ ℓ−
where, putting
eiπµa (ν2) if ν 1
∗
− ≥
c (ν2) = c (ν2) = eiπ(1/2+ℓ µ)a (ν2) if ν < ν 1 (1.13)
∗ ∗ℓ  − ∗ − α ≤ −
e−iπµa∗(ν2) if ν < να,
−
the generalized Dirichlet series Fℓ±(s,α) are defined by
c (ν2) c (ν2)
F+(s,α) = ∗ , F (s,α) = ∗ . (1.14)
ℓ ν 1/2+ℓ ν +ν 2s 1/2 ℓ ℓ− ν 1/2+ℓ ν +ν 2s 1/2 ℓ
α − − α − −
ν>X−να | | | | ν<X−να | | | |
Note that, thanks to the convergence properties of F (s), such generalized Dirichlet series
∗
are absolutely convergent for σ > 1. Note also that the second range in (1.13) is empty if
0 n 1. Finally we define the coefficients a = a (h ) by means of the following polynomial
α ℓ ℓ ∗
≤ ≤
identity
h∗
(X +2j 1) = a (X +ν), (1.15)
ℓ
−
1 j h∗ ℓ=0 0 ν h∗ 1 ℓ
≤Y≤ X ≤ ≤Y− −
where, throughout the paper, an empty product equals 1; for example, a = 1 for every h 0.
0 ∗
≥
With the above notation we have
Theorem. Let α > 0 and ℓ = 0,...,h . Then the functions F (s,α) are entire and F(s,α)
∗ ℓ∗
satisfies the functional equation
ω √N h∗
1 2s
F(s,α) = i√2π 4π − aℓΓ 2(1−s)−1/2−ℓ Fℓ∗(1−s,α). (1.16)
ℓ=0
(cid:0) (cid:1) X (cid:0) (cid:1)
Thus F(s,α) satisfies a functional equation not exactly of Riemann type, but not far from
it. Indeed, one can see from the proofs of Lemma 2.1 and Corollary 1 below that the functions
F (s,α) can be expressed as a kind of stratification of F (s). Therefore, (1.16) is not far
ℓ± ∗
4
from the asymmetric form of the functional equation of F(s), and hence F(s,α) is expected to
behave essentially as a degree 2 function in the extended Selberg class ♯; this is confirmed by
S
the results below. Note also that letting α = 0 in (1.16) one obtains the asymmetric form of
the functional equation of F(s). Indeed, for every ℓ = 0,...,h we have
∗
F (1 s,0) = eiπ(s+µ) e iπ(s+µ) F (1 s),
ℓ∗ − − − ∗ −
and following the initial steps of the proof of the Theorem one can easily rebuild the Γ-factor
(cid:0) (cid:1)
of F(s) from the Γ-factors in (1.16).
Remark 2. The functional equation in the Theorem depends on the special form of the
Γ-factor in (1.6), which enables the explicit computation of a certain hypergeometric function
arising in the proof, see (2.12) and (2.13) below. Actually, similar arguments can be carried
over for a certain class of Γ-factors, for example of type Γ(ds/2+µ)with any d 1 and suitable
≥
real values of µ, thus producing analogous functional equations for the standard twist of the
corresponding L-functions. Here we restrict ourselves to (1.6) for simplicity, since it already
(cid:3)
covers an interesting classical case.
Remark 3. Let α Spec(F). We already pointed out that, in general, F(s,α) has at most
∈
simple poles at the points s = s in (1.8). There are cases where s = s is actually a pole for
ℓ ℓ
infinitely many ℓ’s, see [5] where the case of holomorphic modular forms of integral weight is
treated, and cases where only s = s is a pole, e.g. for F(s) = ζ(s). The existence of infinitely
0
or finitely many poles at the points s = s depends on the structure of certain quotients of Γ
ℓ
functions related to the hypergeometric functions arising as Mellin transform of the Γ-factors in
the functional equation of F(s). In the present case it follows directly from functional equation
(1.16)(butalsofromtheanalysisofsuchΓ-quotients)thatF(s,α)isholomorphicapartpossibly
at s = s with ℓ = 0,...,h , since the functions F (s,α) are entire. An explicit expression for
ℓ ∗ ℓ∗
the residue κ (α) of F(s,α) at s = s , ℓ = 0,...,h , is given in (2.23) below. Moreover, (1.16)
ℓ ℓ ∗
provides an alternative expression for such residues. We finally note that, whatever the value
of α is, the coefficient a (n ) never appears on the right hand side of (1.16) (see (1.12) and
∗ α
(1.14)), but it pops up on the left hand side of (1.16) when α Spec(F), inside the residue at
s = s (see (1.9)). ∈ (cid:3)
0
In the next three subsections we shall introduce several constants, sometimes explicitly de-
pending on α; it is however clear that in general all such constants may depend also on the
function F(s).
1.3. Order of growth. Functional equation (1.16) and the properties of the functions
F (s,α) open the possibility of a further study of the standard twist F(s,α). We already know
ℓ∗
that F(s,α) has polynomial growth on vertical strips, see Theorem 2 of [10], but the bounds
there are definitely weak from a quantitative viewpoint; we denote by
µ(σ) = µ (σ,α)
F
the Lindelo¨f µ-function of F(s,α). The functions F (s,α) have polynomial growth on vertical
ℓ±
strips as well, see Lemma 2.1 below, and for each choice of we denote by
±
µ (σ) = inf ξ : F (σ +it,α) t ξ as t (1.17)
± { 0± ≪ | | → ±∞}
the one-sided Lindelo¨f µ-functionof F (s,α). Onechecks by standardarguments that the main
0±
properties of the Lindelo¨f µ-function of Dirichlet series are satisfied by µ (σ) as well, namely
±
µ (σ) is continuous, convex, non-negative and strictly decreasing until it becomes identically
±
vanishing. Moreover, let
µ (σ) = max µ+(σ),µ (σ) . (1.18)
∗ −
Under the same assumptions of the previous subsection and with the above notation we have
(cid:0) (cid:1)
the following degree 2 convexity bound for F(s,α).
5
Corollary 1. Let α > 0 and ℓ = 0,...,h . Then the functions F (s,α) are entire with
∗ ℓ±
polynomial growth on vertical strips, and
µ(σ) = 1 2σ+µ (1 σ).
∗
− −
Hence, in particular,
µ(σ) = 0 for σ 1 and µ(σ) = 1 2σ for σ 0,
≥ − ≤
and the same holds for µ (σ). Comparing with the bounds for the µ-function of the general
∗
standard twist in [10], we see that Corollary 1 is definitely sharper, of course in the special case
under consideration.
1.4. Trivial zeros. Our last results study the zeros of F(s,α), denoted as usual by ρ =
β+iγ. We first study the trivial zeros, although distinguishing between trivial and non-trivial
zeros is more delicate and unconventional in this case. Indeed, due to the shape of F (s,α) and
ℓ∗
to the lack of Euler product, trivial zeros come from the interferences between the two terms
on the right hand side of (1.12), rather than from the poles of the Γ-factors as in the classical
cases. As it will be clear in a moment, in this paper we restrict ourselves to a rough definition
of trivial zeros, which in principle does not locate them uniquely. Recalling (1.10)-(1.13) let
ν = ν (α) be the value ν > ν for which c (ν2) = 0 and ν +ν is minimum,
+ + α ∗ α
− 6 | |
ν = ν (α) be the value ν < ν for which c (ν2) = 0 and ν +ν is minimum
α ∗ α
− − − 6 | |
and write
c (ν2)
m = m (α) = ν +να and c∗ = √m ∗0 ± = ρ eiθ±; (1.19)
± ± | ± | ± ± ν 1/2 ±
| ±|
clearly, m ,ρ > 0, θ [0,2π) and ρ ,θ depend on α. Further, for ε > 0 let
± ± ± ∈ ± ±
σ m 1 ρ m2
(α) = s C with distance < ε from the line t = log + + log + .
Lε ∈ π m 2π ρ m−2
(cid:26) (cid:18) (cid:19) (cid:18) +(cid:19)(cid:27)
− −
(1.20)
Then the trivial zeros of F(s,α) may be defined as the zeros contained in the part of the
tubular region (α) with σ < σ , with suitable δ > 0 and σ 0. Indeed, under the same
δ − −
L − ≥
assumptions of Subsection 1.2 we have
Corollary 2. Let α > 0. Then there exist δ > 0 and σ 0 such that F(s,α) = 0 for
−
≥ 6
σ < σ unless s (α). More precisely, there exists c (α) > 0 with the property that for
− δ 1
− ∈ L
every 0 < ε < c (α) there exists σ 0 such that
1 ε
≥
i) F(s,α) = 0 for σ < σ unless s (α),
ε ε
6 − ∈ L
ii) there exist infinitely many zeros of F(s,α) in (α) with β < σ .
ε ε
L −
Hence we may choose as σ any fixed value larger than the infimum of the σ ’s, and δ
− ε
accordingly, although in principle this does not define uniquely the trivial zeros. However,
denoting by T (R,α) the number of the zeros in (α) with R β < σ , it is clear from
ε ε ε
L − ≤ −
the proof of Corollary 2 that as R +
→ ∞
T (R,α) = c (α)R+O (1) (1.21)
ε 2 ε
for any given α > 0, with some c (α) = 0 (see in particular (3.8) below). Hence trivial zeros are
2
6
well defined by the above choice, apart from a finite number of them. Note that letting α = 0
we obtain that the line in (1.20) becomes the real axis, since ν = ν and hence m = m
+ +
− − −
and ρ = ρ in this case. Of course, this is consistent with the trivial zeros of F(s), which lie
+
−
on the real axis thanks to the special form of (1.6), and with asymptotic formula (1.21).
Remark 4. Note that when α varies in such a way that √n < ν < √n+1 with some
α
n N, the coefficients of the line in (1.20) may in principle continuously vary between and
∈ −∞
6
+ . But when ν hits the square root of an integer, the position of such a line may change
α
su∞ddenly. This shows an interesting discontinuity in the α-behavior of F(s,α). (cid:3)
We conclude the subsection noting that our present analysis leaves open the problem of the
finer location of the trivial zeros, in particular the determination of a sharp asymptotic region
around the line in (1.20) containing the trivial zeros.
1.5. Non-trivial zeros. Finally we turn to the non-trivial zeros of F(s,α). Let σ+ be the
upper bound of the real parts of the zeros of F(s,α) and σ be as in the previous subsection.
−
The non-trivial zeros of F(s,α) are those in the vertical strip σ σ σ+, thus the zeros of
−
≤ ≤
F(s,α) are the disjoint union of trivial and non-trivial zeros. Let
N (T,α) = ρ = β +iγ : F(ρ,α) = 0,σ β σ+, γ T (1.22)
F −
|{ ≤ ≤ | | ≤ }|
be the counting function of the non-trivial zeros, and let n be the smallest n with a(n) = 0.
0
6
A suitable application of the argument principle gives the following analog of the Riemann-von
Mangoldt formula.
Corollary 3. Let α > 0. Then as T we have
→ ∞
2 T N
N (T,α) = T logT + log + O(logT). (1.23)
F
π π n m m (2πe)2
0 +
−
(cid:0) (cid:1)
Again, note that letting α = 0 in Corollary 3 we get the well known asymptotic formula for
the counting function of the non-trivial zeros of F(s), since m m = n in this case. Also,
+ 0
−
Corollary 3 shows once more the degree 2 behavior of F(s,α). Further, when α varies we can
observe a similar phenomenon as in Remark 4 in the behavior of the coefficient of T in (1.23).
Weconcluderemarkingthat, inadditiontotheinterest comingfromtheapplicationsoutlined
in Subsection 1.1, the above results show that the standard twist F(s,α) is, at least in the case
of half-integral weight modular forms, also a respectable object from the L-functions point of
view. Moreover, although the arguments in this paper are not sufficient to prove a functional
equation for the standard twist of general F ♯, we believe that indeed F(s,α) satisfies a
∈ S
suitable functional equation in such a general case. We shall return on this question in a future
paper, but here we add a final remark.
Remark 5. Thanks to the characterization in [8] of the degree 1 functions of ♯ as linear
S
combinations of Dirichlet L-functions over Dirichlet polynomials from ♯, the standard twists
S
of degree 1 functions in ♯ are closely related to the Hurwitz-Lerch zeta function
S
∞ e( nα)
L(s,α,λ) = − 0 α < 2π, 0 < λ 1.
(n+λ)s ≤ ≤
n=0
X
In particular, one can check that such standard twits satisfy a functional equation of Hurwitz-
Lerch type. Since (1.16) may clearly be regarded as a degree 2 analog of the Hurwitz-Lerch
functional equation, and the same holds in the higher degree cases mentioned in Remark 2, it is
not unreasonable to expect that, in general, standard twists satisfy a kind of general functional
(cid:3)
equation of Hurwitz-Lerch type.
Acknowledgements. We wish to thank Jan Hendrik Bruinier for helpful discussions about
half-integral weight modular forms. This research was partially supported by the Istituto
Nazionale di Alta Matematica and by grants PRIN2015 Number Theory and Arithmetic Ge-
ometry and 2013/11/B/ST1/02799 Analytic Methods in Arithmetic of the National Science
Centre.
7
2. Proof of the Theorem and Corollary 1
2.1. Basic formula. For convenience we write
Q = √N/2π, (2.1)
thus, recalling (1.10), functional equation (1.6) becomes
QsΓ(s+µ)F(s) = ωQ1 sΓ(1 s+µ)F (1 s). (2.2)
− ∗
− −
By the reflection formula Γ(z) 1 = Γ(1 z)sin(πz)/π with z = s + µ, we transform (2.2) in
−
−
the asymmetric form
ωQ1 2s
−
F(s) = Γ(1 s+µ)Γ(1 s µ)sinπ(s+µ)F (1 s).
∗
π − − − −
Therefore, writing for h as in (1.10)
∗
2h +1
∗
µ = (2.3)
∗
4
and observing that µ = µ if h = 0 and µ = µ if h 1, for every h 0 we rewrite the
∗ ∗ ∗ ∗ ∗
± ≥ ≥
above functional equation as
ωQ1 2s
−
F(s) = Γ(1 s+µ )Γ(1 s µ )sinπ(s+µ)F (1 s). (2.4)
∗ ∗ ∗
π − − − −
Thanks to the special value of µ (and hence of µ ) in (1.10) we may suitably transform the two
∗
Γ-factors in (2.4). To this and, since µ h = µ +1/2, we first apply h -times the factorial
∗ ∗ ∗ ∗
− −
formula to the first Γ-factor in (2.4) to get (recall that an empty product equals 1)
Γ(1 s+µ ) = Γ(1 s µ +1/2) (1 s+µ j).
∗ ∗ ∗
− − − − −
1 j h∗
≤Y≤
Then we apply the duplication formula Γ(z)Γ(z +1/2) = π1/221 2zΓ(2z) with z = 1 s µ .
− ∗
− −
In view of (2.3), a simple computation shows that (2.4) becomes
2ω Q
1 2s
F(s) = − Γ(2(1 s) 1/2 h∗)
√2π 2 − − −
(cid:0) (cid:1) (2.5)
(2(1 s) 1/2 h +2j 1) sinπ(s+µ)F (1 s).
∗ ∗
× − − − − −
!
1 j h∗
≤Y≤
Applying identity (1.15) with X = 2(1 s) 1/2 h we see that (2.5) can be written as
∗
− − −
h∗
2ω Q
1 2s
F(s) = − aℓΓ(2(1 s) 1/2 ℓ) sinπ(s+µ)F∗(1 s). (2.6)
√2π 2 − − − ! −
ℓ=0
(cid:0) (cid:1) X
Let now
1
X > 1, α > 0, z = +2πiα (2.7)
X
X
and
∞ a(n) ∞ a(n)
F (s,α) = e( α√n)e √n/X = e zX√n; (2.8)
X ns − − ns −
n=1 n=1
X X
clearly, F (s,α) converges for every s C. Given K > 0, for K < σ < 0 by Mellin’s
X
∈ −
transform we have
1 w
F (s,α) = F(s+ )Γ(w)z wdw,
X 2πi 2 X−
Z(2(K+2))
8
then we shift the line of integration to w = δ; here and later, δ > 0 is a sufficiently small
ℜ
constant, not necessarily the same at each occurrence. Since F(s) is entire, for K < σ < δ
− −
we may use functional equation (2.6) and expand F (1 s w/2) to obtain
∗
− −
h∗
2ω Q 1 2s ∞ a∗(n)
F (s,α) = − a
X √2π 2 n1 s ℓ
−
n=1 ℓ=0 (2.9)
(cid:0) (cid:1) X X
1 w Qz
X w
Γ(2(1 s) 1/2 ℓ w)Γ(w)sinπ(s+ +µ) − dw.
× 2πi − − − − 2 2√n
Z(δ)
(cid:0) (cid:1)
Using the expression sinπz = (eiπz e iπz)/2i we rewrite (2.9) as
−
−
h∗
ω Q 1 2s ∞ a∗(n)
F (s,α) = − a
X i√2π 2 n1 s ℓ (2.10)
−
n=1 ℓ=0
(cid:0) (cid:1) X X
eiπ(s+µ)H(s+ℓ/2,z (n,α)) e iπ(s+µ)H(s+ℓ/2, z (n,α)) ,
X − X
× − −
where
(cid:0) (cid:1)
παQ Q
z (n,α) = i (2.11)
X
√n − 2X√n
in view of (2.7), and
1
H(s,z) = Γ(2(1 s) 1/2 w)Γ(w)z wdw. (2.12)
−
2πi − − −
Z(δ)
An explicit expression for the function H(s,z) can be obtained from the formula for the
Mellin-Barnes integral in (3.3.9) of Ch.3 of Paris-Kaminski [17], namely
1
Γ(ξ w)Γ(w)z wdw = Γ(ξ)(1+z) ξ (2.13)
− −
2πi −
Z(c)
provided 0 < c < ξ and argz < π. To evaluate H(s+ℓ/2, z (n,α)) we have to check the
X
ℜ | | ±
first condition, the second one being certainly satisfied in our case. Recalling that we already
have the condition K < σ < δ, we choose K = h +10 and
∗
− −
2δ < σ < δ if h∗ 2 and sh∗+1 +δ < σ < sh∗ δ if h∗ 3, (2.14)
− − ≤ − ≥
so in view of (1.8) the first condition is also satisfied for every 0 ℓ h . Therefore, from
∗
≤ ≤
(2.12) and (2.13) we get
H(s+ℓ/2, zX(n,α)) = Γ(2(1 s) 1/2 ℓ)(1 zX(n,α))2(s−sℓ), (2.15)
± − − − ±
and inserting (2.15) into (2.10) we obtain, for σ as in (2.14), the basic formula
h∗
ω Q 1 2s ∞ a∗(n)
F (s,α) = − a Γ(2(1 s) 1/2 ℓ)
X ℓ
i√2π 2 − − − n1 s × (2.16)
−
ℓ=9 n=1
(cid:0) (cid:1) X X
eiπ(s+µ)(1+zX(n,α))2(s−sℓ) e−iπ(s+µ)(1 zX(n,α))2(s−sℓ) ,
× − −
where ω, aℓ, zX(n,(cid:8)α) and sℓ are given by (1.6), (1.15), (2.11) and (1.8), resp(cid:9)ecttively. Note
that the series in (2.16) is absolutely convergent since σ < 0 and (1 zX(n,α))2(s−sℓ) 1 as
| ± | →
n .
→ ∞
2.2. Limit as X . Letting X in (2.16) requires some care. Indeed, the limit of
→ ∞ → ∞
F (s,α) is clearly F(s,α) for every α > 0 when σ > 1, but (2.16) holds in a different range,
X
namely for s as in (2.14). Moreover, the limit of the term (1 zX(n,α))2(s−sℓ) is not always well
−
defined, since as X we have 1 z (n,α) 1 παQ/√n, which may vanish for suitable
X
→ ∞ − → −
values of α and n. Actually, in view of (2.1) and of the definition of n and Spec(F) in (1.7),
α
9
such critical values of α and n arise when α Spec(F) and n = n . Therefore, before letting
α
∈
X we derive a different expression for F (s,α).
X
→ ∞
Since we already have information on the analytic properties of F(s,α), see the Introduction
or [10] and [12], we now consider F (s,α) as the twist of F(s,α) by e √n/X, see (2.8). Hence,
X −
by Mellin’s transform, for s as in (2.14), X > 1 and K = h +10 we have
∗
1 w
F (s,α) = F(s+ ,α)Γ(w)Xwdw.
X
2πi 2
Z(2(K+2))
The integrand has simple poles at w = r, r = 0,1,... and, as already mentioned in the
−
Introduction, at most simple poles at w = 2(s s), where s are defined by (1.8). We denote
ℓ ℓ
−
by ρ (α) the residue of F(s+w/2,α) at w = 2(s s); clearly
ℓ ℓ
−
ρ (α) = 2κ (α) (2.17)
ℓ ℓ
where κ (α) is the residue of F(s,α) at s . We shift the line of integration to w = δ
ℓ ℓ
ℜ −
(once more, with a sufficiently small δ > 0, not necessarily the same as at previous places),
thus collecting residues of the poles at w = 0 and, recalling that s is as in (2.14), also at
w = 2(s s) with 0 ℓ h . Therefore, still for s as in (2.14), we get
ℓ ∗
− ≤ ≤
h∗
FX(s,α) = F(s,α)+ ρℓ(α)Γ(2(sℓ s))X2(sℓ−s)
−
ℓ=0
X
1 w (2.18)
+ F(s+ ,α)Γ(w)Xwdw
2πi 2
Z( δ)
−
= F(s,α)+Σ (s,α)+I (s,α),
X X
say. Moreover, since F(s,α) has polynomial growth, see [10], [12] or the Introduction, as
X we have
→ ∞ w
I (s,α) X δ F(s+ ,α)Γ(w)dw 0. (2.19)
X −
≪ | 2 | →
Z( δ)
−
On the other hand, rewriting (2.16) as (recall that a (n ) = 0 if n N)
∗ α α
6∈
h∗
ω Q
1 2s
F (s,α) = − a Γ(2(1 s) 1/2 ℓ)
X ℓ
i√2π 2 − − − ×
ℓ=9
(cid:0) (cid:1) X
× eiπ(s+µ) ∞ an∗1(ns)(1+zX(n,α))2(s−sℓ)
−
n=1
(cid:8) X
−e−iπ(s+µ) ∞ an∗1(ns)(1−zX(n,α))2(s−sℓ) (2.20)
−
n=1
nX=nα
6
a (n )
−e−iπ(s+µ) n∗1 αs (1−zX(nα,α))2(s−sℓ)
α−
= A (s,α) B (s,α) C (s,α) (cid:9)
X X X
− −
say, from (2.18) and (2.20) we get, for X > 1 and s as in (2.14), that
F(s,α) = A (s,α) B (s,α) C (s,α)+Σ (s,α) I (s,α). (2.21)
X X X X X
− − −
From (2.19) and (2.20) it is clear that, as X (cid:0) , I (s,α), A (s,(cid:1)α) and B (s,α) tend, re-
X X X
→ ∞
spectively, to 0 and to well defined functions A(s,α) and B(s,α), say, in the range (2.14). Later
on we shall compute explicitly such functions, but first we treat the critical term C (s,α)+
X
10
Σ (s,α), depending on α belonging to Spec(F) or not. Actually, we show that in both cases
X
C (s,α)+Σ (s,α) vanishes identically.
X X
Suppose first that α Spec(F). Then F(s,α) is entire and a (n ) = 0, hence both C (s,α)
∗ α X
6∈
and Σ (s,α) vanish identically. Suppose now α Spec(F). Then from (1.7), (2.1) and (2.11)
X
∈
we have
Q
1 z (n ,α) = i .
X α
− 2X√n
α
Since in view of (1.8) we have 2(s s) = 2(1 s) 1/2 ℓ, and recalling the definition of
ℓ
− − − −
conductor q of F(s) in (1.3), by a simple computation we obtain
h∗
CX(s,α) = πℓ(α)Γ(2(sℓ s))X2(sℓ−s)
−
ℓ=0
X
with
iωa a (n ) √N
πℓ(α) = −√2πℓ n∗α1−sαℓ 4π 1−2sℓe−iπ(sℓ+µ). (2.22)
Since (s s) > 0 for s as in (2.14), comparing(cid:0) Σ ((cid:1)s,α) in (2.18) with the above expression
ℓ X
ℜ −
for C (s,α)and observing that all other terms in(2.21) remain bounded asX , we deduce
X
→ ∞
that C (s,α)+Σ (s,α) must vanish identically. In particular, thanks to (2.17) and (2.22) the
X X
residue κ (α) of F(s,α) at s = s is explicitly given by
ℓ ℓ
iωa a (n ) √N
κℓ(α) = 2√2ℓπ n∗α1−sαℓ 4π 1−2sℓe−iπ(sℓ+µ) ℓ = 0,...,h∗, (2.23)
(cid:0) (cid:1)
where ω, a and µ are defined by (1.4), (1.15) and (1.10), respectively.
ℓ
Finally we may let X in (2.21) thus getting, for every α > 0 and s as in (2.14), that
→ ∞
F(s,α) = A(s,α) B(s,α). (2.24)
−
2.3. Functional equation. We first compute A(s,α) and B(s,α). Note that by (1.7),
(1.11) and (2.1)
παQ = √n = ν .
α α
Hence, letting X , from (2.11) and (2.20) we obtain at once that
→ ∞
h∗
A(s,α) = i√ω2π Q2 1−2s aℓΓ(2(1−s)−1/2−ℓ)eiπ(s+µ) ∞ an∗1(ns)(1+ √ναn)2(s−sℓ). (2.25)
−
ℓ=9 n=1
(cid:0) (cid:1) X X
From (1.7) and (2.11) we see that for n < n the real part of 1 z (n,α) is negative, hence
α X
−
for such n’s we have
lim (1 zX(n,α))2(s−sℓ) = 1 να 2(s−sℓ)eiπ2(s−sℓ).
X − − √n
→∞ (cid:12) (cid:12)
(cid:12) (cid:12)
Therefore, arguing as for A(s,α) and recalling th(cid:12)at an em(cid:12)pty sum equals 0, we get
(cid:12) (cid:12)
h∗
ω Q
B(s,α) = 1−2s aℓΓ(2(1 s) 1/2 ℓ)e−iπ(s+µ)
i√2π 2 − − − ×
ℓ=0
(cid:0) (cid:1) X (2.26)
eiπ2(s−sℓ) a∗(n) 1 να 2(s−sℓ) + a∗(n) 1 να 2(s−sℓ) .
× n1 s − √n n1 s − √n
( 1≤Xn<nα − (cid:12)(cid:12) (cid:12)(cid:12) nX>nα − (cid:12)(cid:12) (cid:12)(cid:12) )
(cid:12) (cid:12) (cid:12) (cid:12)
Note, as after (2.16), that the series in (2.25) and (2.26) are absolutely convergent since σ < 0.
(cid:12) (cid:12) (cid:12) (cid:12)