The main term

In this section, we will evaluate the contribution of the term (4.48) to (4.44):

X

abe=`

χ(ab)µ(a)τ(b) a√

b

X

w>1

(ae, w) w²

×X

q|c

1 c²

X

h6=0

rw(h)Gχ(h;c)X

±

Z ∞ 0

Lw(±(h−aex))K_(ae,w),w(x)g^±(x,±(h−aex);c)dx.

(4.50)

More precisely, we shall first evaluate thec- andh-sums above then average trivially overa,b,e, w.

To do so we proceed essentially as in [KMV00, pp. 117–122]. We substitute the definition (4.40) ofg^± and make a change of variables

ξ:=|h|

c²y, η := ae

|h|x

in order to remove all parameters from the oscillating functions. Secondly, we replace the negative values of hin (4.50) (which only contribute to the “−” case in P

±) by their absolute values. To simplify the notation, let us write (cf. (4.1))

L(η) :=L_w(hη) = logη+ 2γ+ log h

w²

=: logη+ Λ, say. Then thec, h-sum in (4.50) equals

√1 ae

X

q|c

1 c

X

h>1

rw(h)Gχ(h;c) Z ∞

0

Z ∞ 0

ϕ(4πp ξη)

×n

δη<1L(1−η)J⁺(p

(1−η)ξ) +δη>1L(η−1)J⁻(p

(η−1)ξ) +χ(−1)L(η+ 1)J⁻(p

(η+ 1)ξ)o

×K_(ae,w),w hη

ae

W1

c²ξ hq

W2

abhη eq

dξdη

(ξη)^1/2. (4.51)

Let us also write

Xw(η) :=K_(ae,w),wqη a²b

W2(η).

Its Mellin transformXewsatisfies essentially the same properties asfW2. To see this, observe first that by (4.37),W₂ is up to a negligible error supported on [0, q^ε], so we can replace K_(ae,w),w qη/(a²b) by

K_w^∗(η) :=K_(ae,w),wqη a²b

ω η

q^ε

,

where, as usual,ωis a smooth cut-off function. Then, by (4.6), (4.45), and sufficiently many integra-tions by parts, we find that

Ke_w^∗(u) = Z ∞

0

K_w^∗(η)η^u−1dη_j,<uq^ε|u|^−j for<u >0 and anyj>0. Finally, by (4.21),

Xew(u) = 1 2πi

Z

(¹₂<u)

Ke_w^∗(u−v)fW2(v)dvj,εq^ε|u|^−j forε6<u65, say.

Our next aim is to transform the double integral in (4.51) by several applications of Mellin’s inversion formula: using (4.2) and (2.28), we writeJ^± andϕas inverse Mellin transforms. Then the

ξ, η-integral in (4.51) equals

Since theu1-,u2- andξ-integrals are absolutely convergent (using (4.37)), we can pull theξ-integration inside and calculate it explicitly in terms of the Mellin transform Wf1 of W1. Then we writeXw as an inverse Mellin transform getting that (4.52) equals

Z ∞

Since again all integrals are absolutely convergent, we can pull theη-integration inside and calculate the three terms explicitly (as in [KMV00, (38)]) using [GR07, 3.191.1, 3.191.2, 3.194.3]. We find

Z ∞ it is straightforward to verify that the last two lines in (4.53) can be simplified to

Γ(u3−u4)

In particular, we observe that the triple integral is absolutely convergent (sinceϕ^∗,fW₁, andXe_w are sufficiently nice) and the integrand is holomorphic whenever 0<<u4<<u3<1 +<u1.Let us shift theu4-contour to<u4=ε(<0.1) and theu3-contour to<u3= 1.1.

We now substitute this triple integral back into (4.51) and perform the (absolutely convergent) sum overcandh. To justify this, we need to evaluate fors= 1 + 2u4,t=u3−u4the Dirichlet series

D_w,q(χ, s, t) :=X

q|c

1 c^s

X

h>1

G_χ(h;c)r_w(h) h^t .

First we need to compute the Gauss sum Gχ(h;c): we denote by q^∗ the conductor of χ and, slightly abusing notation, we writeχalso for the primitive character of modulusq^∗underlyingχmod q. For q|c we consider the unique factorizationc =q^∗q1q2c1c2 whereq=q^∗q1q2, c1q1 |(q^∗)^∞ and (c2q2, q^∗) = 1. Then

Gχ(h;c) =χ(c2q2)Gχ(h;q^∗q1c1)rc2q2(h)

(withr_c2q2(h) being the Ramanujan sum, cf. (4.5)). Moreover, G_χ(h;q^∗q₁c₁) = 0 unlessc₁q₁ |hin which case

Gχ(h;q^∗q1c1) =χ h

c1q1

c1q1Gχ(1;q^∗).

Summarizing the above computation, one has Gχ(h;c) =δc₁q₁|hχ(c2q2)χ

h c1q1

c1q1rc₂q₂(h)Gχ(1;q^∗). (4.54) Therefore

D_w,q(χ, s, t) = χ(q₂)G_χ(1;q^∗) (q^∗)^sq₁^s+t−1q^s₂

X

c₁|(q^∗)^∞

1 c^s+t−1₁

X

(c₂,q^∗)=1

χ(c₂) c^s₂

X

h>1

χ(h)r_w(c₁q₁h)r_c2q2(h)

h^t .

For σ := <s, and τ := <t sufficiently large the c1, c2, h-sum factors as an Euler product over the primes:

X

c₁|(q^∗)^∞

1 c^s+t−1₁

X

(c₂,q^∗)=1

χ(c₂) c^s₂

X

h>1

χ(h)r_w(c₁q₁h)r_c2q2(h)

h^t =Y

p

Π_p(χ, s, t), (4.55) say. We collected some useful properties of the Euler factors Πp(χ, s, t) in Lemma 4.1 at the end of this section. These properties imply that for<u4=ε(<0.1) and<u3= 1.1 the series Dw,q(χ, s, t) is absolutely convergent and in the domainσ >1,τ >0 it decomposes as

Dw,q(χ, s, t) =ζ(s+t−1)L(χ, t)Hw,q(χ, s, t), whereH_w,q(χ, s, t) is a holomorphic function. Moreover, for 0< ε <0.1,

<s= 1 + 2ε, ε/2<<t <3ε/2, one has

Hw,q(χ, s, t)εq^ε(q1, w)w^1−ε/3(q^∗)^−1/2. (4.56) Using

Λ = 2γ−log w²

+ log(h) we obtain that (4.51) equals

√1 ae

−4 (2πi)³

Z

(0.2)

Z

(ε)

Z

(1.1)

ϕ^∗(u₁)(2π)^2u4−1Γ(1 +u₁−u₄)Γ(1 +u₁−u₃)Γ(u₃−u₄)

×q^u4fW₁(u₄) ab

eq −u3

Xe_w(u₃)







1

X

j=0

∂^j_u

3 ζ(u₃+u₄)L(ue ₃, u₄) F_j







du₃du₄du₁,

where

L(ue 3, u4) :=L(χ, u3−u4)Hw,q(χ,1 + 2u4, u3−u4) and

F0(u1, u3, u4) :=−πsin(π(u1−u3)) +

cos(π(u1−u3)) +χ(−1) cos(π(u1−u4))

× Γ⁰

Γ(1 +u1−u4)−Γ⁰

Γ(u3−u4) + 2γ−log w²

, F₁(u₁, u₃, u₄) :=−

cos(π(u₁−u₃)) +χ(−1) cos(π(u₁−u₄)) .

Let us now shift the u₃-contour from<u3 = 1.1 to <u3 = 2ε; we will show below that there is no pole atu₃+u₄= 1. Then<(u₃−u₄) =ε, hence

∂_u^j

3L(χ, u₃−u₄)j,ε(q^∗)^1/2

by the functional equation forL(χ, t) with implied constants depending on j, εand (polynomially) on|=(u3−u4)|. In addition, (4.56) combined with Cauchy’s integral formula shows that

∂_u^j₃Hw,q(χ,1 + 2u4, u3−u4)j,εq^ε(q1, w)w^1−ε/3(q^∗)^−1/2, therefore (4.51) summed overabe=`is bounded by

εq^10ε(q1, w) log(w)w^1−ε/3 X

abe=`

(e/ab)^<u3 a√

abe εq^12ε(q1, w)w^1−ε/4`^−1/2.

Finally, averaging over wthe above bound against the weight (ae, w)/w², we obtain that the main term (4.50) is bounded by

εq^13ε`^−1/2. (4.57)

To conclude the analysis of the main term, it remains to show that the pole of the zeta-function atu3+u4= 1 does not contribute anything. Let us only focus on the factors depending onu3:

G(u₁, u₃, u₄) := Γ(1 +u₁−u₃)Γ(u₃−u₄) ab

eq −u3

Xe_w(u₃)







1

X

j=0

∂^j_u

3 ζ(u₃+u₄)eL(u₃, u₄) F_j





 .

IfRj denotes the contribution of thej-term to the residue ofG(u1, u3, u4) atu3= 1−u4, then R₀= Γ(u₁+u₄)Γ(1−2u₄)

ab eq

u4−1

Xe_w(1−u₄)eL(1−u₄, u₄)×

+πsin(π(u₁+u₄))+

+2 sin(πu₁) sin(πu₄)

−2 cos(πu1) cos(πu4) Γ⁰

Γ(1 +u₁−u₄)−Γ⁰

Γ(1−2u₄) + 2γ−log w² , R₁= Γ(u₁+u₄)Γ(1−2u₄)

ab eq

^u4−1

Xe_w(1−u₄)eL(1−u₄, u₄)×

−πsin(π(u₁+u₄))+

+2 sin(πu₁) sin(πu₄)

−2 cos(πu₁) cos(πu₄)

−Γ⁰

Γ(u1+u4) +Γ⁰

Γ(1−2u4) +Xe_w⁰

Xe_w(1−u4)−log ab

eq !

. Here the upper line corresponds toκ= 0 and the lower line toκ= 1, and we have usedχ(−1) = (−1)^κ. Altogether the residual integral equals, after shifting theu₁-integration to (−ε/2) and interchanging theu₁- andu₄-integration,

8 (2πi)²

Z

(ε)

Z

(−ε/2)

ϕ^∗(u₁)(2π)^2u4−1Γ(1 +u₁−u₄)Γ(u₁+u₄)Γ(1−2u₄)

×q^u4Wf₁(u₄) ab

eq ^u4−1

Xe_w(1−u₄)eL(1−u₄, u₄)

×

−sin(πu1) sin(πu4) + cos(πu1) cos(πu4)

Γ⁰

Γ(1 +u₁−u₄)−Γ⁰

Γ(u₁+u₄) +Λ(ue ₄)

du₁du₄,

(4.58)

where

Λ(ue ₄) :=Xe_w⁰

Xe_w(1−u₄) + 2γ−log abw²

eq

. We recast the inner integral as

1 2πi

Z

(−ε/2)

ϕ^∗(u₁)

Λ(ue ₄)−∂_u4

Γ(1 +u₁−u₄)Γ(u₁+u₄)

−sin(πu₁) + cos(πu1)

du₁ and use (2.29) to see that (4.58) equals

8q 2πi

Z

(¹₂)

(2π)^2u4−1Γ(1−2u4)fW1(u4) ab

e ^u4−1

Xew(1−u4)eL(1−u4, u4)

×

sin(πu₄) cos(πu₄)

Λ(ue ₄)−∂_u4 ˆ

ϕ i ¹₂−u₄ ( 1

cot(πu₄) i(¹₂−u4)

) du₄.

Ifϕ=J_k−1,k≡κ (mod 2) then the integral vanishes by ˆϕ= 0. Otherwise we shift∂u₄ to the other factors by partial integration. Then we sum overν as in (4.34) and recall that, by the definition of ϕandW1,

Wf1(u4) =u^2ν₄ Wf(u4) and ϕ(t) = ˆˆ ϕ0(t)αν(t)αξ(t).

Fort∈Rwe have αν(t)∈R, hence the sum overν introduces factors

4

X

ν=0

αν i ¹₂ −u4

u^2ν₄ or

4

X

ν=0

αν i ¹₂−u4 ∂

∂u₄u^2ν₄

to each term. By (4.15)–(4.16) these factors vanish, that is, the residual integral (4.58) is zero in all cases. This completes the analysis of the main term.

Without the additional zeros in the approximate functional equation, we might still succeed at the cost of much more work. Applying the functional equation of L(s, χ), expressing K_(ae,w),w(y) in terms of L ^w

(ae,w)(y) and therefore X_w in terms of W, it should be possible to see that the polar contribution (4.58) resembles exactly the contribution of the cusps a= 0,∞ofQ(`), see (4.29).

We conclude this section by stating and proving some useful properties for the Euler factors Πp(χ, s, t) in (4.55).

Lemma 4.1. Let σ=<s >1andτ=<t >0. For a primepletvp denote thep-adic valuation, and letζp (resp. Lp) denote the corresponding Euler factor of the Riemann zeta function (resp. Dirichlet L-function).

a) For(p, qw) = 1,

Πp(χ, s, t) =ζp(s+t−1)Lp(χ, t) L_p(χ, s). b) For p|q^∗,

|Πp(χ, s, t)|63p^{min(vp(q1),vp}(w))+(1−τ)v_p(w)ζp(σ−1)ζp(τ).

c) For (p, q^∗) = 1,p|qw,

|Πp(χ, s, t)|64p^{vp(q2)+(1−τ)vp(w)}ζp(σ−1)ζp(τ).

Proof. a) For (p, qw) = 1 we use the notation

α:=vp(c2), β :=vp(h)

in the sum (4.55), then in the sum (4.55), then clearly

|Πp(χ, s, t)|6 We distinguish between two cases. Forγ>δwe infer

|Π_p(χ, s, t)|6

In both cases we conclude

|Πp(χ, s, t)|63pmin(γ,δ)+δ(1−τ)ζp(σ−1)ζp(τ).

c) For (p, q^∗) = 1,p|qw, we use the notation

α:=vp(c2), β:=vp(h), γ:=vp(q2), δ:=vp(w) in the sum (4.55), then clearly

|Πp(χ, s, t)|6

We distinguish between two cases. Forγ>δwe infer (note thatγ >0 in this case)

In both cases we conclude

|Πp(χ, s, t)|64p^{γ+δ(1−τ)}ζp(σ−1)ζp(τ).

The proof of Lemma 4.1 is complete.

In document Subconvex Bounds for Automorphic L-functions and Applications (Pldal 51-57)