Hypergeometric Function

25 Hypergeometric function

IMRN International Mathematics Research NoticesVolume 2006, Article ID 41417, Pages 1–19

Integral Mean Values of Maass L-Functions

Qiao Zhang

1 Introduction

A standing topic in analytic number theory is to estimate integral mean values

∫T

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2m

dt, (1.1)

where f is an automorphic form over GL(n) and L(s, f) its associated L-function, nor-

malized so that the central point is at s = 1/2. Over the last ninety years, many au-

thors have worked in this field with fruitful results; see, for example, an excellent sur-

vey in [7]. In most cases, we have to face delicate discussions of the arithmetic nature

of Fourier coefficients, such as estimates of the “generalized additive divisor problems”∑n≤x λf(n)λf(n + r).

To avoid this difficulty, in this paper we consider instead the Dirichlet integral,

Zf(w) =

∫∞

1

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

t−wdt (�w� 1), (1.2)

and study its analytic properties, especially its meromorphic continuation beyond �w =

1 and its polar behavior. Our goal is to express Zf(w) as an inner product of fwith a cer-

tain kernel function, by directly exploring the symmetries satisfied by f itself, so that

Received 11 April 2005; Accepted 19 February 2006

Communicated by Dennis Hejhal

at Shandong University on M

arch 29, 2014http://im

rn.oxfordjournals.org/D

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http://imrn.oxfordjournals.org/


2 Qiao Zhang

it suffices to study the analytic properties of this kernel function alone. This realizes

and generalizes the ideas suggested by Good [3], and is applicable to cases where our

knowledge of the arithmetic nature of f is still limited. This is the main motivation of the

present work.

Along this line, in [9] we have considered modular forms with respect to Hecke

congruence subgroups, and studied the mean squares of their L-functions. The present

paper is devoted to the study of the corresponding results for Maass forms. In particular,

we have the following theorem.

Theorem 1.1. Let f(z) be an even Maass form for Γ0(N) with Laplacian eigenvalue 1/4+ν2

and nebentypus χ(modN). Then asymptotically,

∫T

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

dt ∼

8

Γ

(1

2+ iν

)

Γ

(1

2− iν

)‖f‖2

vol(Γ\H)· T log T

=8 coshπν

π

‖f‖2

vol(Γ\H)· T log T,

(1.3)

where the norm of f is given by

‖f‖2 =

∫∫

Γ0(N)\H

∣∣f(z)

∣∣2dxdy

y2. (1.4)

�

Remark 1.2. The mean squares of the L-functions associated to Maass forms were firstly

studied in 1981 by Kuznetsov [6], who used the method of approximate functional equa-

tions to obtain the asymptotic formula (1.3) for Maass forms with respect to SL(2,Z).

Along this line, in 1992 Muller considered nonholomorphic automorphic forms over

Fuchsian groups of the first kind with real weight. In 1997, Jutila [5] used the Laplace

transform method to study in a unified way the fourth moment of ζ(s) and the mean

squares of L-functions associated to both cusp forms and Maass forms with respect to

SL(2,Z). However, all these cases rely on delicate discussions of certain generalized ad-

ditive divisor problems.

Remark 1.3. As in [9], we can also obtain a much more accurate asymptotic formula for a

weighted mean value problem, but we will not give details here.

Remark 1.4. Using the renormalized integrals as introduced by Zagier [8], we may also

expect a corresponding result along this line for the fourth moment of the Riemann zeta




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Integral Mean Values of Maass L-Functions 3

function or a Dirichlet L-function. However, due to the lack of space, we omit the discus-

sion here.

The arguments for Theorem 1.1 are essentially the same as those in [9], and the

main obstacle here is to establish an inner product representation for the Dirichlet in-

tegral (1.2), as now general hypergeometric functions, instead of the gamma functions,

come into the picture. Similar difficulties arise in the recent work of Beineke and Bump

[1] in which the authors take yet another approach to study the mean squares but get only

the upper bound

∫T

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

dt� T log T, (1.5)

where f is an even Maass form with respect to SL(2,Z).

To better illustrate the idea of our approach, now we present the proof for

Theorem 1.1 but leave the verification of some crucial estimates to the later sections.

Proof for Theorem 1.1. For simplicity, in this proof we write Γ = Γ0(N).

Write

f(z) =

∞∑n=−∞n�=0

a(n)√yKiν

(

2π|n|y)

e2πinx, (1.6)

then for �s > 1we have

∫∞

0

f(

reiθ)

rs−1/2dr

r=√

sin θ∞∑

n=−∞n�=0

an

∫∞

0

Kiν

(

2π|n|r sin θ)

e2πinr cos θrsdr

r

=2√

sin θ(2π sin θ)s

L(s, f)∫∞

0

Kiν(r) cos(r cot θ)rsdr

r.

(1.7)

By the integral formula (2.4), this gives that

∫∞

0

f(

reiθ)

rs−1/2dr

r=1

2(sin θ)1/2−sΛ(s, f) 2F1

(s + iν

2,s − iν

2;1

2; −(cot θ)2

)

, (1.8)




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4 Qiao Zhang

where

Λ(s, f) = π−sΓ

(s + iν

2

)

Γ

(s − iν

2

)

L(s, f) (1.9)

is the complete L-function for f, so by (2.2) we have the Mellin transform

∫∞

0

f(

reiθ)

rs−1/2dr

r=1

2(sin θ)1/2+iνΛ(s, f) 2F1

(s + iν

2,1 − s + iν

2;1

2; (cos θ)2

)

.

(1.10)

Hence the inverse Mellin transform gives

f(

reiθ)

=(sin θ)1/2+iν

4πi

∫(2)Λ(s, f) 2F1

(s + iν

2,1 − s + iν

2;1

2; (cos θ)2

)

r1/2−sds,

(1.11)

where we write∫

(c) to denote the integral∫c+i∞

c−i∞ .

Also, for �τ > 1 and �w ≥ 1 let

Pw,τ(z) =∑γ∈Γ

(�γz)τ

(�γz

|γz|

)w

(1.12)

be the nonholomorphic kernel function, then in [9] we have shown that Pw,τ(z) has a

meromorphic continuation up to �τ > −ε, �w > 1/2. In particular, Pw,0(z) is holomor-

phic up to �w = 1with a double pole atw = 1 and Laurent expansion

Pw,0(z) =4

vol(Γ\H)1

(w − 1)2+O

(1

|w − 1|

)

. (1.13)

Now assume that �τ > 1 and �w ≥ 1, and consider the inner product

Zf(w, τ) =⟨

Pw,τ, |f|2⟩

=

∫∫Γ\H

Pw,τ(z)∣∣f(z)

∣∣2dxdy

y2

=

∫∞

0

∫∞

−∞yτ

(y

|z|

)w∣∣f(z)

∣∣2dxdy

y2,

(1.14)

then the above analytic properties of Pw,τ(z) imply that Zf(w, τ) also has a meromorphic

continuation up to �τ > −ε, �w > 1/2, and that Zf(w, 0) is holomorphic up to �w = 1




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with a double pole atw = 1 and Laurent expansion

Zf(w, 0) =4

vol(Γ\H)‖f‖2

(w − 1)2+O

(1

|w − 1|

)

. (1.15)

On the other hand, we may further assume that �τ > 2, then using the polar

coordinates in H and the inverse Mellin transform (1.11) gives

Zf(w, τ) =

∫∞

0

∫π

0

rτ(sin θ)w+τ−2f(

reiθ)

f(

reiθ)

dθdr

r

=1

4πi

∫(2)Λ(s, f)ds

∫π

0

2F1

(s + iν

2,1 − s + iν

2;1

2; (cos θ)2

)

(sin θ)3/2−w−τ−iνdθ

·∫∞

0

rτ+1/2−sf(

reiθ)dr

r,

(1.16)

and another application of the Mellin transform (1.10) now gives

Zf(w, τ) =1

4πi

∫(2)Λ(s, f)Λ(τ + 1 − s, f)I(s;w, τ)ds, (1.17)

where the function

I(s;w, τ)

=1

2

∫π

0

2F1

(s+iν

2,1−s+iν

2;1

2; (cos θ)2

)

2F1

(τ+1−s−iν

2,s−τ−iν

2;1

2; (cos θ)2

)

(sin θ)1−w−τdθ

=

∫π/2

0

2F1

(s+iν

2,1−s+iν

2;1

2; (cos θ)2

)

2F1

(τ+1−s−iν

2,s−τ−iν

2;1

2; (cos θ)2

)

(sin θ)1−w−τdθ

(1.18)

obviously has an analytic continuation up to �s ≥ 1/2, �(w + τ) ≥ 1. In particular, for

w > 1we have

Zf(w, 0) =1

4πi

∫(2)Λ(s, f)Λ(1 − s, f)I(s;w, 0)ds

=1

4πi

∫(1/2)

Λ(s, f)Λ(1 − s, f)I(s;w, 0)ds

=1

2π

∫∞

0

∣∣∣∣Λ

(1

2+ it, f

)∣∣∣∣

2

I

(1

2+ it;w, 0

)

dt.

(1.19)




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6 Qiao Zhang

Now we quote the estimates (3.5) and (3.28), to be proved later in Propositions 3.2 and

3.3 in Section 3, that for t → ∞ we have

I

(1

2+ it;w, 0

)

=

2w−1πΓ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)∣∣∣∣Γ

(1

4+it + iν

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4+it − iν

2

)∣∣∣∣

2t−w +O

(

eπtt1/2−w)

,

(1.20)

where theO-constant is uniform forw ∈ [1, 3/2]. Plugging this into (1.19) gives

Zf(w, 0) =

Γ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)2w−2

π

∫∞

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

t−wdt +G(w)

(1.21)

for some function G(w) analytic for �w > 1/2.

Finally, we combine this formula with (1.15), then asw → 1we have

∫∞

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

t−wdt ∼

24−wπΓ(w)

Γ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

‖f‖2

vol(Γ\H)1

(w − 1)2

∼

8π

Γ

(1

2

)2

Γ

(1

2+ iν

)

Γ

(1

2− iν

)

‖f‖2

vol(Γ\H)1

(w − 1)2

=8

Γ

(1

2+ iν

)

Γ

(1

2− iν

)‖f‖2

vol(Γ\H)1

(w − 1)2.

(1.22)

Hence by classical complex Tauberian arguments we have

∫T

0

∣∣∣∣L

(1

2+ it, f

)∣∣∣∣

2

dt ∼

8

Γ

(1

2+ iν

)

Γ

(1

2− iν

)‖f‖2

vol(Γ\H)· T log T

=8 coshπν

π

‖f‖2

vol(Γ\H)· T log T.

(1.23)




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Note that we have applied the well-known formula

Γ

(1

2+ z

)

Γ

(1

2− z

)

= π sec(πz). (1.24)

This completes the proof of Theorem 1.1. �

2 Hypergeometric functions

For reference, in this section we summarize some results on hypergeometric functions

that we will use in sequel.

Let a, b, c ∈ C, then the hypergeometric function 2F1(a, b; c; z) is defined by the

infinite series,

2F1(a, b; c; z) =

∞∑n=0

Γ(a + n)Γ(a)

Γ(b + n)Γ(b)

Γ(c)Γ(c + n)

zn

n!

(

|z| < 1)

, (2.1)

with meromorphic continuation in z to the whole complex plane.

Now we quote some well-known basic formulas for hypergeometric functions.

Lemma 2.1. There exists

2F1(a, b; c; z) = (1 − z)−a2F1

(

a, c − b; c;z

z − 1

)

.

(2.2)

�

Lemma 2.2. Assume that 1 − c, b − a, and c − a − b are not integers. Then

2F1(a, b; c; z) =Γ(c)Γ(c − a − b)Γ(c − a)Γ(c − b) 2F1(a, b;a + b − c + 1; 1 − z)

+Γ(c)Γ(a + b − c)

Γ(a)Γ(b)2F1(c − a, c − b; c − a − b + 1; 1 − z)

(1 − z)a+b−c.

(2.3)

�

Lemma 2.3. Assume that �a = 0 and that �s > |�ν|. Then

∫∞

0

Kν(x) cos(ax)xsdx

x= 2s−2Γ

(s + ν

2

)

Γ

(s − ν

2

)

2F1

(s + ν

2,s − ν

2;1

2; −a2

)

.

(2.4)�




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8 Qiao Zhang

The main ingredient of this paper is the following asymptotic estimate for hyper-

geometric functions.

Lemma 2.4 (see [4]). Assume that |z| ≤ 1 but z = −1, then as |λ| → ∞

2F1

(

a + λ, b − λ; c;1 − z

2

)

= Γ(c)2(a+b−1)/2 (z + 1)(c−a−b−1)/2

(z − 1)c/2

(sinh ζζ

)1/2

α1−c(

ζIc−1(αζ) + ε(ζ))

,

(2.5)

where we have put z = cosh ζwith �ζ ≥ 0 and |�ζ| ≤ π, and

α =a − b

2+ λ. (2.6)

Furthermore, the error term ε(ζ) is bounded by

ε(ζ) � |ζ|

|α|2∣∣Kc(αζ)

∣∣, (2.7)

where the �-constant depends on a, b, c only. �

Lemma 2.5 (see [4]). In the notation of the above lemma, the error term ε(ζ) can be ob-

tained as follows. Consider the sequence of functions {εn(ζ)} with ε0(ζ) = 0,

ε1(ζ) = 2ζ

∫ζ

0

tIc−1(αt)(

Ic−1(αζ)Kc−1(αt) − Ic−1(αt)Kc−1(αζ)){tB0(t)

} ′dt,

εn+1(ζ)−εn(ζ)=ζ

∫ζ

0

(

Ic−1(αζ)Kc−1(αt)−Ic−1(αt)Kc−1(αζ))(

εn(t)−εn−1(t))

ψ(t)dt,

(2.8)

where

B0(ζ) =1

2ζ

((

(c − 1)2 −1

4

)(1

ζ− coth ζ

)

+(c − 1)2 − (a + b − c)2

2tanh

ζ

2

)

,

ψ(ζ) =

(

(c − 1)2 −1

4

)(1

sinh2ζ

−1

ζ2

)

+(c − 1)2 − (a + b − c)2

4 cosh2 ζ

2

.

(2.9)




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Then

ε(ζ) = limn→∞ εn(ζ) =

∞∑n=0

(

εn+1(ζ) − εn(ζ))

. (2.10)�

3 Propositions and their proofs

Now we are ready to prove the estimate (1.20) of I(1/2 + it;w, 0) that we quoted in the

proof of Theorem 1.1.

Proposition 3.1. Letw ∈ [1, 3/2] and write s = 1/2 + it, then as t → ∞,

∫π/2

log t/2t

2F1

(s + iν

2,1 − s + iν

2;1

2; (cos θ)2

)

2F1

(s − iν

2,1 − s − iν

2;1

2; (cos θ)2

)

(sin θ)1−wdθ

� eπtt1/2−w,

(3.1)

where the �-constant is uniform inw. �

Proof. Applying the estimate (2.5) with z = − cos 2θ and recalling the well-known for-

mula

I−1/2(x) =

√

2

πxcosh x, (3.2)

we have, as t → ∞, that

2F1

(1

4+iν

2+it

2,1

4+iν

2−it

2;1

2; (cos θ)2

)

∼

et(π/2−θ) + e−t(π/2−θ)

2(sin θ)1/2+iν, (3.3)

so forw ≥ 1 the integral in (3.1) is asymptotically equal to

∫π/2

log t/2t

(sin θ)w−2

(et(π/2−θ) + e−t(π/2−θ)

2

)2

dθ

� eπt

∫π/2

log t/2t

θw−2e−2θtdθ� eπtt1−w

∫∞

(1/2) log t

θw−2e−2θdθ

� eπtt1−w

∫∞

(1/2) log t

e−θdθ� eπtt1/2−w.

(3.4)

This completes the proof. �




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10 Qiao Zhang

To prove the estimate (1.20), we consider the cases ν = 0 and ν = 0 separately, as

the latter requires some additional treatment.

Proposition 3.2. Letw ∈ [1, 3/2]. Assume that ν = 0, then

I

(1

2+ it;w, 0

)

=

2w−1πΓ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)∣∣∣∣Γ

(1

4+it + iν

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4+it − iν

2

)∣∣∣∣

2t−w +O

(

eπtt1/2−w)

,

(3.5)�

where the �-constant is uniform inw.

Proof. The integral over [log t/2t, π/2] has been estimated in Proposition 3.1. For 0 ≤ θ ≤log t/2t, since ν = 0, by (2.3) we have

2F1

(1

4+iν

2+it

2,1

4+iν

2−it

2;1

2; (cos θ)2

)

=

Γ

(1

2

)

Γ(−iν) 2F1

(1

4+iν

2+it

2,1

4+iν

2−it

2; 1 + iν; (sin θ)2

)

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

+

Γ

(1

2

)

Γ(iν) 2F1

(1

4−iν

2+it

2,1

4−iν

2−it

2; 1 − iν; (sin θ)2

)

(sin θ)2iνΓ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

)

=

√π

(sin θ)iν

∑ε=±1

Γ(−iεν) 2F1

(1

4+iεν

2+it

2,1

4+iεν

2−it

2; 1 + iεν; (sin θ)2

)

(sin θ)−iενΓ

(1

4−iεν

2+it

2

)

Γ

(1

4−iεν

2−it

2

)

=

√π

(sin θ)iνF(t; θ),

(3.6)

say, then by symmetry we also have

2F1

(1

4−iν

2+it

2,1

4−iν

2−it

2;1

2; (cos θ)2

)

=√π(sin θ)iνF(t; θ), (3.7)




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so

I

(1

2+ it;w, 0

)

= π

∫ log t/2t

0

(sin θ)w−1F(t; θ)2dθ +O(

eπtt1/2−w)

. (3.8)

Applying (2.5) with z = cos 2θ, we have

(sin θ)iν

Γ(−iν) 2F1

(1

4+iν

2+it

2,1

4+iν

2−it

2; 1 + iν; (sin θ)2

)

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

=νΓ(iν)Γ(−iν)

2Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

(2

t

)iν2iθIiν(tθ) + ε(2iθ)

(θ sin θ)1/2.

(3.9)

Now (2.7), together with the well-known asymptotic formula

Ks(x) ∼

√π

2xe−x (x −→ ∞), (3.10)

shows that the error term ε(2iθ) is bounded by

ε(2iθ) � θ

t21

∣∣K1+iν(tθ)

∣∣� θ

t2

(

1 +√tθetθ

) � θtε−3/2, (3.11)

so Stirling’s formula gives

(sin θ)iν

Γ(−iν) 2F1

(1

4+iν

2+it

2,1

4+iν

2−it

2; 1 + iν; (sin θ)2

)

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

=

iνΓ(iν)Γ(−iν)(2

t

)iν

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

(θ

sin θ

)1/2

Iiν(tθ) +O(

e(π/2)ttε−1)

.

(3.12)




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12 Qiao Zhang

By symmetry, we also have

(sin θ)−iν

Γ(iν) 2F1

(1

4−iν

2+it

2,1

4−iν

2−it

2; 1 − iν; (sin θ)2

)

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

)

= −

iνΓ(iν)Γ(−iν)(2

t

)−iν

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

)

(θ

sin θ

)1/2

I−iν(tθ) +O(

e(π/2)ttε−1)

,

(3.13)

so

F(t; θ) =

(θ

sin θ

)1/2 ∑ε=±1

iενΓ(iν)Γ(−iν)(2

t

)iεν

Iiεν(tθ)

Γ

(1

4−iεν

2+it

2

)

Γ

(1

4−iεν

2−it

2

) +O(

e(π/2)ttε−1)

.

(3.14)

As we recall that

I−s(z) = Is(z) +2 sin(sπ)

πKs(z), (3.15)

the above implies that

F(t; θ) = iνΓ(iν)Γ(−iν)(

θ

sin θ

)1/2

Iiν(tθ)∑

ε=±1

ε

(2

t

)iεν

Γ

(1

4−iεν

2+it

2

)

Γ

(1

4−iεν

2−it

2

)

+

(θ

sin θ

)1/2 2ν sinh(πν)Γ(iν)Γ(−iν)(2

t

)−iν

Kiν(tθ)

πΓ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

) +O(

e(π/2)ttε−1)

= iνΓ(iν)Γ(−iν)(

θ

sin θ

)1/2

Iiν(tθ)∑

ε=±1

ε

(2

t

)iεν

Γ

(1

4−iεν

2+it

2

)

Γ

(1

4−iεν

2−it

2

)

+

(θ

sin θ

)1/2 2

(2

t

)−iν

Kiν(tθ)

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

) +O(

e(π/2)ttε−1)

,

(3.16)




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where we have applied the well-known formula

Γ(iν)Γ(−iν) =π

ν sinh(πν). (3.17)

By Stirling’s formula, we have (see, e.g., [2, formula 1.18.4])

Γ(z + α)Γ(z + β)

= zα−β

(

1 +(α − β)(α + β − 1)

2z+O

(

z−2))

(−π < arg z < π), (3.18)

so

(2

t

)iν

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

) −

(2

t

)−iν

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

)

=

(2

t

)iν

⎛

⎜⎜⎝1 −

(2

t

)−2iν Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

)

⎞

⎟⎟⎠

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

� t−2−�(iν)∣∣∣∣Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)∣∣∣∣

� e(π/2)tt−3/2.

(3.19)

It is also well known that, for θ ∈ [0, log t/2t],

Iiν(tθ) � min

{1,etθ

√tθ

}� t1/2

√

log t� t1/2, (3.20)

so we have

F(t; θ) =

(θ

sin θ

)1/2 2

(2

t

)−iν

Kiν(tθ)

Γ

(1

4+iν

2+it

2

)

Γ

(1

4+iν

2−it

2

) +O(

eπ/2tε−1) � e(π/2)tt1/2.

(3.21)




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14 Qiao Zhang

By symmetry, we also have

F(t; θ) =

(θ

sin θ

)1/2 2

(2

t

)iν

Kiν(tθ)

Γ

(1

4−iν

2+it

2

)

Γ

(1

4−iν

2−it

2

)

+O(

eπ/2tε−1) � e(π/2)tt1/2.

(3.22)

Therefore

∫ log t/2t

0

(sin θ)w−1F(t; θ)2dθ

=4

∣∣∣∣Γ

(1

4+iν

2+it

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4−iν

2+it

2

)∣∣∣∣

2

∫ log t/2t

0

(sin θ)w−2θKiν(tθ)2dθ

+O

(

eπttε−1/2

∫ log t/2t

0

(sin θ)w−1dθ

)

=4

∣∣∣∣Γ

(1

4+iν

2+it

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4−iν

2+it

2

)∣∣∣∣

2

∫ log t/2t

0


+O(

eπtt−w)

.

(3.23)

Furthermore, we have

∫ log t/2t

0


=

(

1 +O

((log t)2

t2

)) ∫ log t/2t

0

θw−1Kiν(tθ)2dθ

= t−w

(

1 +O

((log t)2

t2

)) ∫∞

0

θw−1Kiν(θ)2dθ +O(1)

=

2w−3Γ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)t−w +O(1),

(3.24)




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where we have applied the well-known formula

∫∞

0

Ka(y)Kb(y)ysd×y =

Γ

(s + a + b

2

)

Γ

(s + a − b

2

)

Γ

(s − a + b

2

)

Γ

(s − a − b

2

)

23−sΓ(s),

(3.25)

so

∫ log t/2t

0

(sin θ)w−1F(t; θ)2dθ

=

2w−1Γ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)∣∣∣∣Γ

(1

4+iν

2+it

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4−iν

2+it

2

)∣∣∣∣

2t−w +O

(

eπtt−w)

.

(3.26)

Finally, we combine the above estimate with Proposition 3.1, and this gives

I

(1

2+ it;w, 0

)

= π

∫ log t/2t

0

(sin θ)w−1F(t; θ)2dθ +O(

eπtt1/2−w)

=

2w−1πΓ

(w

2

)2

Γ

(w

2+ iν

)

Γ

(w

2− iν

)

Γ(w)∣∣∣∣Γ

(1

4+iν

2+it

2

)∣∣∣∣

2∣∣∣∣Γ

(1

4−iν

2+it

2

)∣∣∣∣

2t−w +O

(

eπtt1/2−w)

.

(3.27)


Proposition 3.3. Letw ∈ [1, 3/2] and write s = 1/2 + it. Assume that ν = 0, then

I

(1

2+ it;w, 0

)

=

2w−1πΓ

(w

2

)4

Γ(w)∣∣∣∣Γ

(1

4+it

2

)∣∣∣∣

4t−w +O

(

eπtt1/2−w)

, (3.28)

where the �-constant is uniform inw. �




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16 Qiao Zhang

Proof. The integral over [log t/2t, π/2] has been estimated in Proposition 3.1. For 0 ≤ θ ≤log t/2t, we may apply (2.3) by taking the limit as ν → 0, and this gives

2F1

(1

4+it

2,1

4−it

2;1

2; (cos θ)2

)

= −

Γ

(1

2

)(

2γ +Γ ′

Γ

(1

4+it

2

)

+Γ ′

Γ

(1

4−it

2

))

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

Γ

(1

4+it

2

)

Γ

(1

4−it

2

)

−

Γ

(1

2

)(

2 log sin θ +∂

∂a+∂

∂b+ 2

∂

∂c

)

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

Γ

(1

4+it

2

)

Γ

(1

4−it

2

) ,

(3.29)

where ∂/∂a, ∂/∂b, and ∂/∂c denote the partial derivatives with respect to the first, the

second, and the third parameters of the hypergeometric function, respectively.

As before, the asymptotic formula (2.5) gives

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)(θ

sin θ

)1/2

I0(tθ) +O(

t−3/2) � t1/2. (3.30)

Also, Stirling’s formula implies that

Γ ′

Γ(s) = log s +O

(

|s|−1)

, (3.31)

so the above formula can be simplified as

2F1

(1

4+it

2,1

4−it

2;1

2; (cos θ)2

)

= −

Γ

(1

2

)(

2γ + 2 logt

2+ 2 log sin θ

)

Γ

(1

4+it

2

)

Γ

(1

4−it

2

)

(θ

sin θ

)1/2

I0(tθ)

−

Γ

(1

2

)(∂

∂a+∂

∂b+ 2

∂

∂c

)

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

Γ

(1

4+it

2

)

Γ

(1

4−it

2

) +O(

e(π/2)t).

(3.32)




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As for the partial derivatives, by (2.5) we have

∂

∂a2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

=(θ sin θ)−1/2

2i

(

− log cos θ(

2iθI0(tθ) + ε(2iθ))

− 2θ2I ′0(tθ) +∂

∂aε(2iθ)

)

,

∂

∂b2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

=(θ sin θ)−1/2

2i

(

− log cos θ(

2iθI0(tθ) + ε(2iθ))

+ 2θ2I ′0(tθ) +∂

∂bε(2iθ)

)

,

∂

∂c2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

=

(Γ ′

Γ(1) + log cos θ − log sin θ − log

t

2

)2iθI0(tθ) + ε(2iθ)2i(θ sin θ)1/2

+1

2i(θ sin θ)1/2

(

2iθ∂I0(tθ)∂c

+∂

∂cε(2iθ)

)

=

(

− γ + log cos θ − log sin θ − logt

2

)2iθI0(tθ) + ε(2iθ)2i(θ sin θ)1/2

−

(θ

sin θ

)1/2

K0(tθ) +1

2i(θ sin θ)1/2

∂

∂cε(2iθ).

(3.33)

Note that here we have applied the formula that

∂

∂cIc(x)

∣∣c=0

= −K0(x). (3.34)

Hence

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

=2√π

Γ

(1

4+it

2

)

Γ

(1

4−it

2

)

(θ

sin θ

)1/2

K0(tθ) +O(

e(π/2)ttε−1)

+O

(

e(π/2)tt1/2θ−1

(∣∣∣∣

∂

∂aε(2iθ)

∣∣∣∣+

∣∣∣∣

∂

∂bε(2iθ)

∣∣∣∣+

∣∣∣∣

∂

∂cε(2iθ)

∣∣∣∣

))

.

(3.35)




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18 Qiao Zhang

To estimate the above partial derivatives for ε(2iθ), we can in fact follow the ar-

guments in [4] for Lemma 2.5 line by line with only minor modifications. The discussion

is lengthy but routinic, so we omit the details here but only state the estimates

∂

∂aε(2iθ) � θ

t2K1(tθ)� θtε−3/2,

∂

∂bε(2iθ) � θ

t2K1(tθ)� θtε−3/2,

∂

∂cε(2iθ) � θ

(

1 +∣∣ log(tθ)

∣∣)

t2K1(tθ)� θtε−3/2| log θ|.

(3.36)

Hence

2F1

(1

4+it

2,1

4−it

2; 1; (sin θ)2

)

=2√π

∣∣∣∣Γ

(1

4+it

2

)∣∣∣∣

2

(θ

sin θ

)1/2

K0(tθ) +O(

e(π/2)ttε−1| log θ|) � e(π/2)tt1/2.

(3.37)

Combining the above estimate with Proposition 3.1, as in the proof of Proposition 3.2 we

have

I

(1

2+ it,w; 0

)

=

∫ log t/2t

0

(sin θ)w−12F1

(1

4+it

2;1

2; (cos θ)2

)2

dθ +O(

eπtt1/2−w)

=4π

∣∣∣∣Γ

(1

4+it

2

)∣∣∣∣

4

∫ log t/2t

0

(sin θ)w−2θK0(tθ)2dθ +O(

eπtt1/2−w)

=4π

∣∣∣∣Γ

(1

4+it

2

)∣∣∣∣

4t−w

∫∞

0

θw−1K0(θ)2dθ +O(

eπtt1/2−w)

=

2w−1πΓ

(w

2

)4

Γ(w)∣∣∣∣Γ

(1

4+it

2

)∣∣∣∣

4t−w +O

(

eπtt1/2−w)

.

(3.38)





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References

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nal of Number Theory 105 (2004), no. 1, 175–191.

[2] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions.

Vols. I, II, McGraw-Hill, New York, 1953, based in part, on notes left by Harry Bateman.

[3] A. Good, The convolution method for Dirichlet series, The Selberg Trace Formula and Related

Topics (Brunswick, Maine, 1984), Contemporary Mathematics, vol. 53, American Mathematical

Society, Rhode Island, 1986, pp. 207–214.

[4] D. S. Jones, Asymptotics of the hypergeometric function, Mathematical Methods in the Applied

Sciences 24 (2001), no. 6, 369–389.

[5] M. Jutila, Mean values of Dirichlet series via Laplace transforms, Analytic Number Theory,

London Mathematical Society, Lecture Note Series, vol. 247, Cambridge University Press, Cam-

bridge, 1997, pp. 169–207.

[6] N. V. Kuznetsov, Mean value of the Hecke series of a cusp form of weight zero, Differential ge-

ometry, Lie groups and mechanics, IV, Zapiski Nauchnykh Seminarov Leningradskogo Otde-

leniya Matematicheskogo Instituta 109 (1981), 93–130, 181, 183.

[7] K. Matsumoto, Recent developments in the mean square theory of the Riemann zeta and other

zeta-functions, Number Theory, Trends Math., Birkhauser, Basel, 2000, pp. 241–286.

[8] D. Zagier,The Rankin-Selberg method for automorphic functions which are not of rapid decay,

Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 28 (1981), no. 3,

415–437 (1982).

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no. 1, 100–122.

Qiao Zhang: Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

E-mail address: [email protected]




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