The infinite place ∞ ofk corresponds to the degree valuation

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L-FUNCTIONS OVER RATIONAL FUNCTION FIELDS

CHIH-YUN CHUANG

Abstract. The aim of this paper is to establish asymptotic formulas in the region

<(s)≥1of the`-th moment of quadraticL-functions over a rational function fieldF^q(t), for arbitrary positive integer`and odd prime powerq. Specifically, we obtain the asymptotic formulas relating to two families. One is over all discriminants and another is over all fundamental discriminants. In addition, we give applications including asymptotic formulas for the size of class numbers, algebraic K-groupsK2, and limit distribution functions associated with the quadraticL-functions.

Introduction

Let k = Fq(t) be a rational function field with odd characteristic p. Let A = Fq[t], and A⁺ be the set of all monic polynomials. The infinite place ∞ ofk corresponds to the degree valuation. The letter P always denotes a monic irreducible polynomial in A, which corresponds to a finite place ofk.

Let_·

·

be the quadratic symbol forFq[t](cf. [9, Chapter 3]). Given m∈A non-square, define the function χm(n) := m

n

for n ∈ A− {0}. We are interested in the L-function associated toχm which is defined by

L(s, χm) := X

n∈A⁺

χm(n)q^−sdegⁿ =Y

P

(1−χm(P)q^−sdeg^P)⁻¹, on<(s)>1.

This is equal to a polynomial of degree at mostdegm−1in q^−s. Suppose thatm∈Ais square-free. Let

λ_∞(m) :=







0, if∞is ramified ink(√ m)/k;

−1, if∞is inert ink(√ m)/k;

1, if∞splits ink(√ m)/k.

Then

L^∗(s, χ_m) := (1−λ_∞(m)q^−s)⁻¹·L(s, χ_m)

is the Artin L-function associated to the unique non-trivial character of the Galois group Gal(k(√

m)/k) (cf. [9, Theorem 17.6]). Let m(χm) := degm−1− |λ∞(m)|. Then the functional equation for the completeL-functionL^∗(s, χm)is as follows:

L^∗(s, χm) =q^m(χ^m^)(1/2−s)L^∗(1−s, χm),

which implies thatL(s, χ_m)is a polynomial of degreedegm−1inq^−s. We classify quadratic function fieldsK:=k(√

m)according to whether∞splits, is inert, or ramified inK/k. This is analogous to classifying quadratic number fields as real or imaginary. That is

• Ifdegmis even andsgn(m)∈(F^×q )², then∞splits inK/k.

• Ifdegmis even, andsgn(m)6∈(F^×q)², then∞is inert inK/k.

2010Mathematics Subject Classification. 14H05, 11F67, 11N45, 11L05.

Key words and phrases. Function fields,L-functions, Asymptotic Results, Gauss sums.

Research partially supported by Ministry of Science and Technology, Rep. of China.

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• Ifdegmis odd, then∞ramifies inK/k.

Heresgn(m)is the leading coefficient of m∈A.LetB be the integral closure ofA in K/k.

Then theL-functionL(s, χm)satisfies

ζ_B(s) =ζ_A(s)·L(s, χ_m), whereζA ( reps. ζB) is the zeta function ofA( reps. B):

• ζA(s) := X

I⊂A

N(I)^−s = Y

P

(1−q^−s^deg^P)⁻¹ on <(s) > 1, where N(I) denotes the absolute norm.

• ζ_B(s) := X

I⊂B

N(I)^−s =Y

P

(1−N(P)^−s)⁻¹ on <(s)>1, where the sum is over all non-zero ideals inB, and the product is over all non-zero prime ideals inB.

These zeta functions both have a simple pole ats= 1, and are rational functions inq^−s. Throughout, we use the symbolto denote square polynomials. Letγbe a fixed generator ofF^×q, and `be a positive integer. On the basis of classifying the quadratic fieldsK overk, we are interested in considering the following averaging problems:

• Summing over all non-square monic polynomials (“discriminants”):

L(s, M, `)R= X

m∈A+ : degm=M

L(s, χm)^`, ifM is an odd integer;

L(s, M, `)_S = X

m∈A+ : degm=M,m6=

L(s, χm)^`, ifM is an even integer;

L(s, M, `)_I = X

m∈A+ : degm=M,m6=

L(s, χ_γ·m)^`, ifM is an even integer.

• Summing over all square-free monic polynomials (“fundamental discriminants”):

L^∗(s, M, `)_R:= P∗

m∈A+ : degm=M

L(s, χm)^`, ifM is an odd integer;

L^∗(s, M, `)_S := P∗

m∈A+ : degm=M

L(s, χm)^`, ifM is an even integer;

L^∗(s, M, `)_I:= P∗

m∈A+ : degm=M

L(s, χ_γ·m)^`, ifM is an even integer.

Here∗means that the sum in question runs over all square-free monic polynomials.

We note that for any odd integerM, L(s, M, `)R= P

m∈A+ : degm=M

L(s, χ_γ·m)^`, andL^∗(s, M, `)_R = P∗

m∈A+ : degm=M

L(s, χ_γ·m)^`.

We divide our results into two parts.

0.1. The non-square case: In this case, when ` = 1, J. Hoffstein and M. Rosen (cf. [6, Theorem 0.7, Theorem 1.4 and Theorem 1.5]) obtained formulas for above sums, as functions ofq^−s, which immediately gives asymptotic formulas asM approaches infinity. One of goals of this paper is derived asymptotic formulas for these sums whenM goes to infinity with` arbitrary. That is

Theorem 0.1. Let `, M be positive integers and ? be either S, I, orR, then we have, for

<(s)≥1,

L(s, M, `)?=ζ_A(2s)^`(`+1)² ·c_`(s)·q^M+O

q^(1/2+δ)M ,

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for anyδ >0, asM → ∞. Herec_`(s) :=

Y

P

(1−q⁻^deg^P)

(1 +q^−sdeg^P)^`+ (1−q^−s^deg^P)^` 2

+q⁻^deg^P(1−q^−2s^deg^P)^`

(1−q^−2s^deg^P)^`(`−1)²

is absolutely convergent on<(s)>1/4.

Remark 0.2.

(1). When`= 1,

c₁(s) =ζ_A(2s+ 1)⁻¹

is a rational function inq^−s. The above theorem reduces to the well-known averaging L-values of J. Hoffstein and M. Rosen (cf. [9, p. 323-324]).

(2). LetDdenote an integer congruent to0or1modulo4and non-square. LetψD(n) :=

D n

denote the Kronecker symbol. The DirichletL-function associated with ψD is given by

L(s, ψ_D) :=

∞

X

n=1

ψ_D(n)/n^s, on<(s)>1.

Concerning higher moments, M. B. Barban [2, Lemma 5.6] established the following asymptotic formula, for any fixed positive integer`,

X

−N≤D≤−1

L^`(1, ψD) =r`N+O(Nexp(−c√

N)), as N→ ∞, wherec >0 is a constant independent of`and

r`=

∞

X

n=1 n≡1 mod 2

ϕ(n)d`(n²) n³ .

Hered`(n) is the number of ways of expressingnas the product`positive integers, expressions in which only the order of the factors begin different is regarded as distinct, andϕ(n)is the Euler totient function. In the function field case, the constant of our main term is (cf. (2.1)):

ζA(2s)^`(`+1)² ·c`(s) = X

n∈A⁺

d`(n²)·ϕ(n)

|n|^2s+1 :=r`(s).

Here|n|:=q^degⁿ for anyn∈A− {0}. In particular, ifs= 1, then ζA(2)^`(`+1)² ·c`(1) = X

n∈A⁺

d`(n²)·ϕ(n)

|n|³ ,

which is to be compared with the classical result. Hered`(n)is the number of ways of expressingnas the product`monic polynomials, expressions in which only the order of the factors begin different is regarded as distinct, andϕ(n) is the Euler totient function forA.

(3). Althoughζ_A^`(s) = X

n∈A⁺

d`(n)

|n|^s and ^ζ^A_ζ^(s−1)

A(s) = X

n∈A⁺

ϕ(n)

|n|^s are both rational functions inq^−s, we do not know whetherζA(2s)^`(`+1)² ·c`(s) = X

n∈A⁺

d`(n²)·ϕ(n)

|n|^2s+1 is a rational function inq^−sfor` >1.

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(4). For each local factor at primeP, we have

(1−q⁻^deg^P)

(1 +q^−sdeg^P)^`+ (1−q^−s^deg^P)^` 2

=1 + X

t:2|t 2≤t≤`

` t

q^−st^deg^P−q⁻^deg^P X

t:2|t 2≤t≤`

` t

q^−st^deg^P+q⁻^deg^P

`

X

t=1

` t

(−1)^tq^−2st^deg^P,

which is a polynomial of degree2`in q^−s^deg^P. For each` >1, the infinite product Y

P

(1−q⁻^deg^P)

(1 +q^−sdeg^P)^`+ (1−q^−s^deg^P)^` 2

is holomorphic on <(s) > 1/2 and has a pole at s = 1/2 of order ^`(`−1)₂ . Since c_`(s)is equal toζ_A(2s)⁻^`(`−1)² multiplying this infinite product, the functionc_`(s)is holomorphic on<(s)>1/4.

In M. B. Barban’s paper, he not only obtains the asymptotic formula of higher moment described in the above remark (2), but also achieves a result of limit distributions with its corresponding characteristic function (cf. [2, Theorem 5.2]). In the function field context, we also have an analogous result and prove it in Subsection 2.1:

Corollary 0.3. For real s0≥1, asM → ∞, the quantity

fM(x, s0) =q^−M#{m∈A⁺,degm=M : L(s0, χm)≤x}, x∈R

converges to a distribution functionf(x) :=f(x, s₀)at each point of continuity of the latter, and the corresponding characteristic function has the form

φf,s₀(x) = 1 +X

`≥1

r`(s0)

`! (ix)^`, x∈R.

Writem=m0m²₁, wherem0 is square-free. The polynomialm0 is well defined up to the square of a constant. Define Bm to be the ring A+Am1

√m0 ⊂ K = k(√

m). It is an A-order in K, (i.e. it is a ring, finitely generated as an A-module, and its quotient field is K). Meanwhile, B_mis the unique subring of B_m₀ such thatm₁ is the annihilator of the A moduleBm₀/Bm(cf. [9, Theorem 17.6]). The Picard groupPic(Bm)is the group of invertible fractional ideals ofBm modulo the subgroup of principal fractional ideals. We set the class numberhm:= # Pic(Bm).

Since L(1, χm) gives the size of Pic(Bm), setting s = 1 in Theorem 0.1, we obtain the following average value results for the`-th moment ofh_m.

Corollary 0.4. Let ` be a positive integer.

(1). Ifdegm=M is an odd integer, then X

m∈A+ : degm=M,m6=

h^`_m=q(¹⁺^`₂)^M·q^−`/2·ζ_A(2)^`(`+1)² ·c_`(1) +O(q(^1/2+δ+₂^`)^M).

(2). Ifdegm=M is an even integer andγ is a generator ofF^×q, then X

m∈A+ : degm=γM,m6=

h^`_γm=q(¹⁺2^`)^M · 2

q+ 1 ^`/2

·ζA(2)^`(`+1)² ·c`(1) +O(q(^1/2+δ+^`2)^M).

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(3). Ifdegm=M is an even integer, then X

m∈A+ : degm=M,m6=

(hm·Rm)^`=q(¹⁺2^`)^M ·(q−1)^−`/2·ζA(2)^`(`+1)² ·c`(1) +O(q(^1/2+δ+2^`)^M).

HereRmis the regulator of the ringBm.

0.2. The square-free case: When ` = 1 and q ≡ 1 mod 4, J. Hoffstein and M. Rosen established that (cf. [6, Theorem 0.8]):

Theorem 0.5. Let P(s) :=Y

P

1− |P|⁻²− |P|^−(2s+1)+|P|^−(2s+2)

, on<(s)≥1/2.

Choose any >0. If<(s)≥1, then (1). IfM is odd, then

(q−1)⁻¹(q^M−q^M−1)⁻¹X

L(s, χm) =ζA(2)ζA(2s)P(s) +O(q^{−(M/2)(1−)}), where the sum is over all square-freemsuch that deg(m) =M.

(2). IfM is even, then

2(q−1)⁻¹(q^M−q^M⁻¹)⁻¹X

L(s, χ_m) =ζ_A(2)ζ_A(2s)P(s) +O(q^{−(M/2)(1−)}), where the sum is over all square-freem such that deg(m) =M andsgn(m)∈(F^×q)² or over all square-freem such thatdeg(m) =M andsgn(m)6∈(F^×q )².

Hoffstein-Rosen uses the fact that the Fourier coefficients of Eisenstein series for the meta- plectic group involve the values L(s, χ_m). In this paper, we are able to derive asymptotic formulas for these averaging values, withq, `arbitrary, by more elementary direct approach.

We have

Theorem 0.6. Let `, M be positive integers. Suppose that? is eitherS,I, orR. Then, for

<(s)≥1,

L^∗(s, M, `)?=ζA(2s)^`(`+1)² ·c^∗_`(s)·(q^M ·ζA(2)⁻¹) +O

q^(1/2+δ)M for anyδ >0, asM → ∞. Here

c^∗_`(s) :=Y

P







1−q^−2s^deg^P^`(`+1)₂ 1 +q⁻^deg^P

(1 +q^−s^deg^P)^−`+ (1−q^−s^deg^P)^−`

2

+q⁻^deg^P





 ,

which is absolutely convergent in<(s)>1/4. Here

(q^M ·ζ_A(2)⁻¹) =q^M−q^M−1= #{m∈A⁺: degm=M andmis square-free}, forM >1.

Remark 0.7.

(1). When`= 1, we have (cf. (2.3)) ζA(2s)^`(`+1)² ·c^∗₁(s) =ζA(2)·ζA(2s)·Y

P

(1− |P|⁻²− |P|^−2s−1+|P|^−2s−2) :=r^∗_`(s).

The above theorem reduces to the averagingL-values of J. Hoffstein and M. Rosen (cf. [6, Theorem 5.2]) in the region<(s)≥1.

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(2). For each local factor at primeP, we have





1−q^−2sdeg^P^`(`+1)₂ 1 +q⁻^deg^P

(1 +q^−sdeg^P)^−`+ (1−q^−sdeg^P)^−`

2

+q⁻^deg^P



(1−q^{−2 deg}^P)

=

1 + `(`−1)

2 q^−2s^deg^P+

`

X

t=4 2|t

` t

q^−st^deg^P−

`

X

t=2 2|t

` t

q−(st+1) degP

+

`

X

t=1

(−1)^t `

t

q−(2st+1) degP

−

`

X

t=0

(−1)^t `

t

q−(2st+2) degP

· 1−q^−2s^deg^P

`(`−1) 2

which is a polynomial of degree`(`+ 1)inq^−s^deg^P. Thus, for each`≥1, the infinite product gives thatc^∗_`(s)is holomorphic on<(s)>1/4.

Similarly, Theorem 0.5 also gives a limit distribution function of the square-free case.

Corollary 0.8. For real s0≥1, asM → ∞, the quantity

f_M^∗(x, s0) = (q^M−q^M−1)⁻¹#{m∈A⁺,degm=M andmis square-free: L(s0, χm)≤x}, x∈R converges to a distribution functionf^∗(x) :=f^∗(x, s₀)at each point of continuity of the latter, and the corresponding characteristic function has the form

φ^∗_f,s₀(x) = 1 +X

`≥1

r_`^∗(s0)

`! (ix)^`, x∈R.

Use the same argument as Corollary 0.4. Setting s = 1 in Theorem 0.6, we obtain the following average value results for the`-th moment ofh_m.

Corollary 0.9. Let ` be a positive integer. Then, for any δ >0, (1). Ifdegm=M is an odd integer, then

P∗

m∈A+ : degm=M

h^`_m=q(¹⁺^`₂)^M·q^−`/2·ζ_A(2)⁻¹·ζ_A(2)^`(`+1)² ·c^∗_`(1) +O(q(^1/2+δ+^`₂)^M).

(2). Ifdegm=M is an even integer, and γ is a generator ofF^×q, then P∗

m∈A+ : degm=M

h^`_γm=q(¹⁺^`₂)^M ·

2 q+1

^`/2

·ζA(2)⁻¹·ζA(2)^`(`+1)² ·c^∗_`(1) +O(q(^1/2+δ+₂^`)^M).

(3). Ifdegm=M is an even integer, then P∗

m∈A+ : degm=M

(hm·Rm)^`=q(¹⁺^`2)^M·(q−1)^−`/2·ζA(2)⁻¹·ζA(2)^`(`+1)² ·c^∗_`(1) +O(q(^1/2+δ+^`2)^M).

HereRmis the regulator of the ringBm.

Remark 0.10. LetF be a quadratic extension overQ. Let∆_{F /}_Q,hF, andRF be the discrim- inant ofF/Q, the class number, and the regulator ofF, respectively. In 2008, T. Taniguchi conjectured that (cf.[11, Theorem 1 and Conjecture 10.14]):

X→∞lim 1

X² · X

[F:Q]=2 0<|^∆F /Q|^≤X

h²_F·R²_F = ζ(2)² 2⁴ ·Y

p

1− 3

p³+ 2 p⁴ + 1

p⁵ − 1 p⁶

.

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When`= 2, simplifying our cases(1),(2), and(3)in the above corollary, we have proved (1) + (2) + (3)

=q(¹⁺^`₂)^M· 1

q−1 + 2 q+ 1+1

q

·ζA(2)⁻¹·ζA(2)^`(`+1)² ·c^∗₂(1)

= 1

q−1+ 2 q+ 1+1

q

·ζA(2)²·Y

P

1− 3

|P|³ + 2

|P|⁴+ 1

|P|⁵ − 1

|P|⁶

·q^2M,

which is to be compared with T. Taniguchi’s conjecture.

SinceL(2, χm)gives the size of K-groupsK2(Bm)(after Tate and Quillen, cf. [8]), setting s= 2in Theorem 0.6, we also obtain the following average value results for the`-th moment of#(K2(Bm))for arbitrary`.

Corollary 0.11. Let `be a positive integer, m∈A be square-free. Then, for anyδ >0, (1). Ifdegm=M is an odd integer, then

P∗

m∈A+ : degm=M

(#K2(Bm))^`=q⁻^3`²·ζA(2)⁻¹·ζA(4)^`(`+1)² ·c^∗_`(2)·q(¹⁺^3`2)^M+O(q(1/2+δ+3`/2)M).

(2). Ifdegm=M is an even integer, and γ is a generator ofF^×q, then P∗

m∈A+ : degm=M

(#K2(Bγ·m))^`=_(1+q−1) q²+1

^`

·ζA(2)⁻¹·ζA(4)^`(`+1)² ·c^∗_`(2)·q(¹⁺^3`₂)^M +O(q(1/2+δ+3`/2)M).

(3). Ifdegm=M is an even integer, then P∗

m∈A+ : degm=M

(#K2(Bm))^`= (q²+q)^−`·ζA(2)⁻¹·ζA(4)^`(`+1)² ·c^∗_`(2)·q(¹⁺^3`2)^M +O(q(1/2+δ+3`/2)M).

The strategy of proving Theorem 0.1 and Theorem 0.6 is similar. We rewriteL(s, M, `)?

(orL^∗(s, M, `)_?) as a polynomial inq^−swhose coefficients involve general divisor functiond_` as well as quadratic Gauss sums (cf. Lemma 3.1 and Lemma 4.1). For the sake of applying a function field version of the Tauberian theorem, we divide the above sums into parts according to the locations and orders of poles ofL-functions associated tod`and quadratic Gauss sums (cf. Corollary 2.6). Finally, we mention that the main contribution comes form the quadratic Gauss sums degenerating to the character sums (cf. Proposition 3.2 and Proposition 4.2).

The contents of this paper are as follows. In subsection 1.1, we introduce the quadratic Gauss sums in our context and derive their exact values. The `-th moment of L-functions associated with quadratic character χm for each positive integer ` is studied in subsection 1.2. We then establish asymptotic formulas in section 2 by a function field version of the Tauberian theorem. Proofs of Corollary 0.3 and Corollary 0.8 are given in subsection 2.1.

We finally prove Theorem 0.1 in section 3 and Theorem 0.6 in section 4.

1. Preliminaries

Letk_∞be the completion field ofkat∞,O_∞ be the valuation ring ofk_∞, andπ_∞=t⁻¹ be a fixed uniformizer. Fory :=

∞

X

i=N

a_iπ_∞ⁱ ∈k_∞ witha_N 6= 0, we define ord_∞(y) :=N. We

fix an additive characterψ_∞ofk_∞ as the following, fory:=

∞

X

i=N

aiπⁱ_∞∈k_∞,

ψ_∞(y) :=exp(2πi

p Tr_F_q_/_F_p(−a1))∈C^×.

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For a locally constant function with compact supportf :k_∞→C, the Fourier transformf^∗ off is defined to be

fˆ(y) :=

ˆ

k∞

f(x)ψ_∞(xy)dx,

wheredx is the Haar measure of k_∞ such that Vol(O_∞) =q. Thenfˆˆ(x) =f(−x)holds. It is straightforward to prove that:

Proposition 1.1. (Poisson summation formula)Letf be a locally constantC-valued function with compact support. Then for anyx∈k^×_∞ andy∈k_∞, we have

X

m∈A

f(xm+y) =q^ord^∞^(x) X

m⁰∈A

fˆ m⁰

x

·ψ_∞

−ym⁰ x

.

1.1. Quadratic symbol and Gauss sums. Let _a

b

be the Kronecker symbol forA with a∈ A and b ∈ A⁺. For a ∈A, a 6= 0, we define sgn₂(a)to be the leading coefficient of a raised to the ^q−1₂ power. The reciprocity law of this symbol is stated as follows:

Proposition 1.2. (The quadratic reciprocity law) Let a, b∈A be relatively prime, nonzero elements. Then

a b

b a

= (−1)^q−1² ^deg^a^deg^bsgn₂(a)^deg^bsgn₂(b)⁻^deg^a.

Letn be a monic polynomial. For all polynomialse∈A, we define an analogue of Gauss sum as follows:

Ge(n) := X

a modn

ha n i

ψ∞

−ae n

∈C,

and put

G˜e(n) :=

1 +i

2 +

−1 n

1−i 2

Ge(n).

(1.1)

Herei:=√

−1.Before we compute the exact values ofG˜_e(n)for alln∈A⁺, we note that Lemma 1.3. Let P be a fixed monic irreducible polynomial. For all integers N satisfying l_deg_P

2

m≤N ≤degP−1, we have

X

n∈A degn=N

ψ_∞

−n² P

= 0.

Proof. Letn=

0

X

i=−N

aiπⁱ_∞ andP⁻¹=

∞

X

j=degP

bjπ^j_∞, where a−N, bdegP ∈F^×q, and ai, bj∈Fq

for all−N+ 1≤i≤0 andj≥degP+ 1. Then

n² P =

∞

X

u=−2N+degP





 X

j≥degP

−N≤i1,i2≤0 i1 +i2 +j=u

ai1ai2bj





 π^u_∞.

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Therefore, we have

X

n∈A degn=N

ψ∞

−n² P

= X

a−N∈F× q al∈Fq ,−N <l≤0

ψ∞







−





 X

j≥degP

−N≤i1,i2≤0 i1 +i2 +j=1

ai₁ai₂bj





 π∞







= X

Y

j≥degP

−N≤i1,i2≤0 i1 +i2 +j=1

ψ_∞((−a_i₁a_i₂b_j)π_∞).

The definition of ψ∞ implies that we only focus on i1+i2+j = 1. Taking i1 = −N and j= degP in the above equality, we havei₂= 1−degP+N,which implies that

−degP 2

+ 1≤i₂≤0

by the assumption ofN. Thus, there always existsi₂6=−N such that−N+i₂+ degP = 1.

Meanwhile, for thisi2 = 1−degP +N, the solutioni1 =−N and j = degP is the unique solution satisfyingi1+i2+j= 1for−N ≤i1≤0 anddegP ≤j. Thus, we derive

X

n∈A degn=N

ψ_∞

−n² P

= X







Y

j≥degP

−N≤i1,i2≤0 i1 +i2 +j=1

(i1,i2,j)6=(−N,1−degP+N,degP) (i1,i2,j)6=(1−degP+N,−N,degP)

ψ_∞((−ai₁ai₂bj)π_∞)







·ψ_∞(−2·a_−N ·bdegP·a_1−degP+N ·π_∞)

= X

a−N∈F× q al∈Fq ,−N <l6=1−degP+N≤0







Y

j≥degP

−N≤i1,i2≤0 i1 +i2 +j=1

(i1,i2,j)6=(−N,1−degP+N,degP) (i1,i2,j)6=(1−degP+N,−N,degP)

ψ∞((−ai₁ai₂bj)π∞)







·





X

a1−degP+N∈Fq

ψ_∞(a_1−degP+N ·π_∞)



= 0.

Now, we computeG˜e(n)for alln∈A⁺.

Lemma 1.4. (1). Supposemandnare co-prime monic polynomials. Then G˜e(mn) = ˜Ge(m) ˜Ge(n).

(2). Suppose thatd∈A, andαis the largest power of irreducible polynomialP dividinge (Ife= 0 then set α=∞). Then forβ ≥1

G˜e(P^β) :=











0, ifβ ≤αis odd;

ϕ(P^β), ifβ ≤αis even;

−q^α^deg^P, ifβ =α+ 1 is even;

(γp,q)^deg^P ·h

eP^−α P

i·q(α+1/2) degP, ifβ =α+ 1 is odd;

0, ifβ ≥α+ 2.

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Here,

γp,q :=− − s

−1 p

!^[Fq:Fp]

· 1 +i

2 + (−1)^q−1² 1−i 2

∈ {±1}.

(1.2)

where

· p

is the Legendre symbol modulop,[Fq:Fp]is the dimension ofFq overFp, and for f ∈A,ϕ(f) := #(A/f A)^× is the Euler totient function forA.

Proof. The statement (1). comes from the Chinese Remainder theorem and the quadratic reciprocity law.

(2). We only show the crucial case β = α+ 1. Others are straightforward to verify. If β=α+ 1, then

G_e(P^β) = X

a modP^β

h a P^β

iψ_∞

−ae P^β

= X

l modP

l P^β

X

b modP^β−1

ψ∞

−(bp+l)e P^β

=q^{(β−1) deg}^P X

l modP

l P^β

ψ∞

−le P^β

.

Ifβ is even, then X

l modP

l P^β

ψ_∞

−le P^β

=−1. Setdeg 0 =−∞. Ifβ is odd, and degP is odd, then

X

l modP

l P

ψ_∞

−l(eP^−α) P

= eP^−α

P

X

l modP

ψ_∞

−l² P

= eP^−α

P







X

l modP 2 degl−degP <−1

ψ_∞

−l² P

+ X

l modP 2 degl−degP=−1

ψ_∞

−l² P

+ X

l modP 2 degl−degP >−1

ψ_∞

−l² P





, by Lemma 1.3

= eP^−α

P



q^(deg^P−1)/2+q^(deg^P−1)/2 X

a∈F^×q

exp

2πiTr_F_q_/_F_p(a²) p





= eP^−α

P

q^(deg^P−1)/2 X

a∈Fq

exp

2πiTr_F_q_/_F_p(a²) p

=− − s−1

p

!^[^F^q^:^F^p^]eP^−α P

q^deg^{P /2}, by Davenport-Hasse relation [7, p. 158-162].

Meanwhile, in this case, we have ¹⁺ⁱ₂ +₋₁

P^β

_1−i

2 =¹⁺ⁱ₂ + (−1)^q−1² ¹⁻ⁱ₂ .Combining the above equalities, we obtain the desired result.

Ifβ=α+ 1is odd anddegP is even, then we have X

l modP

l P

ψ_∞

−l(eP^−α) P

= eP^−α

P

q(β−1/2) degP

by the same method as the above case.

1.2. Quadratic L-functions. Let K := k(√

m) be a quadratic field over k, where m is non-square withdegm≥1. One of our goals is to investigate the mean value of`-th moment of the class numbershm. If∞doesn’t split inK/k, then B_m^× =F^×q, and if∞splits inK/k, thenB_m^× =F^×q×< m>, where< m>is infinite cyclic. In this case, we set Rm equal to the absolute value oflog_qq^ord^∞⁽^m⁾.

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Supposem is square-free, the connection betweenL(1, χ_m) and class numbers is proven by E. Artin. This result also can be generalized to the case of non-square polynomials (cf.

[9, Theorem 17.8B]) .

Theorem 1.5. Let m∈Abe a non-square polynomial of degree M ≥1.

(1). L(1, χm) =q^1−M² ·hm, ifM is odd.

(2). L(1, χm) = ^q+1₂ ·q^−M/2·hm, if M is even and sgn₂(m) =−1.

(3). L(1, χ_m) = (q−1)·q^−M/2·h_m·R_m, ifM is even and sgn₂(m) = 1. HereR_m is the regulator of the ringB_m.

Suppose thatmis square-free. We are able to investigate the mean value of`-th moment of#(K2(Bm)), since the connection betweenL(2, χm)and#(K2(Bm))is already known by Tate and Quillen. That is (cf. [8, Proposition 2]):

Theorem 1.6. Let m∈Abe a square-free polynomial of degree M ≥1. Then (1). #(K₂(B_m)) =q^(3/2)M·q^−3/2·L(2, χ_m), ifM is odd.

(2). #(K₂(B_m)) =q^(3/2)M·(1+q⁻¹)·(q²+1)⁻¹·L(2, χ_m), ifM is even andsgn₂(m) =−1.

(3). #(K2(Bm)) =q^(3/2)M·(q²+q)⁻¹·L(2, χm), if M is even and sgn₂(m) = 1.

For each positive integer`, the`-th moment ofL-functionL(s, χm)is:

L(s, χm)^`

=







M−1

X

N=0





 X

n∈A+ : degn=N

χm(n)





q^−sN







`

=

`(M−1)

X

N=0





 X

n∈A+ : degn=N

d`(n)χm(n)





q^−sN,

where

d`(n) := X

n1,n2,...,n`∈A+ : n1n2···n`=n

1 (1.3)

is the number of ways of expressingnas the product ofkmonic polynomials, expressions in which only the order of the factors being different is regarded as distinct. The main purpose of this paper is to study the mean values of these`-th moments.

2. Asymptotic formulas for arithmetic functions

The following Tauberian Theorem is used to study asymptotic formulas for arithmetic functions (cf. [3, Theorem 7]):

Theorem 2.1. Letf(u) := X

N≥0

aNu^N with the numbers aN ∈C for allN, be convergent in

{u∈C:|u|< q^−a}

for a fixed real numbera >0. Assume that in the above domain f(u) =g(u)(u−q^−a)^−w+h(u)

holds, whereh(u),g(u)are analytic functions in{u∈C:|u| ≤q^−a},g(q^−a)6= 0, and w >0 is a positive integer. Then

a_N = (−1)^−wg(q^−a)q^aw

Γ(w) ·q^aNN^w−1+O q^aNN^w−2

, asN → ∞.

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Leta= 1, then Theorem 2.1 reduces to the case [9, Theorem 17.4].

Corollary 2.2. Letf(u) := X

N≥0

a_Nu^N with the numbersa_N ∈Cfor allN, be convergent in

{u∈C:|u|< q^−a}

for a fixed real numbera >0. Assume that in the above domain f(u) =g(u)(u+q^−a)^−w+h(u)

holds, whereh(u),g(u)are analytic functions in{u∈C:|u| ≤q^−a},g(−q^−a)6= 0 andw is a positive integer. Then

aN = (−1)^Ng(−q^−a)q^aw

Γ(w) q^aNN^w−1+O q^aNN^w−2

, asN → ∞.

Proof. Set f˜(u) =f(−u), ˜g(u) =g(−u)and ˜h(u) =h(−u). Thus, f(u) :=˜ X

N≥0

(−1)^NaNu^N The conditionf(u) =g(u)(u+q^−a)^−w+h(u)is equivalent to

f˜(u) = (−1)^−wg(u)(u˜ −q^−a)^−w+ ˜h(u).

Applying Theorem 2.1, we obtain that

(−1)^NaN =g(−q^−a)q^aw

Γ(w) q^aNN^w−1+O q^aNN^w−2 .

Let`be a positive integer. The arithmetic functiond`(n²)will also play a key role in our asymptotic studies. Hered`(n)is the general divisor function defined in (1.3). Let ϕ(n)be the Euler totient function forA. Applying Theorem 2.1 witha= 3 andw=^`(`+1)₂ , it is not difficult to drive the estimation below

Lemma 2.3. Let `be a positive integer, then

X

n∈A+ : degn=N

d`(n²)ϕ(n²) =c`(1/2)·q^3N ·N^`(`+1)² ⁻¹ Γ_`(`+1)

2

+O q^3N·N^`(`+1)² ⁻¹ N

! ,

asN→ ∞.

Proof. For`≥1, we note that

∞

X

t=0

(2t+ 1)(2t+ 2)·...·(2t+`−1) (`−1) ! x^t=

(1 +√

x)^`+ (1−√ x)^` 2

(1−x)^−`.

Suppose`= 1. d`(n) = 1for alln∈A⁺. Thus, X

n∈A+ degn=N

d`(n²)ϕ(n²) =q^3N(1−q⁻¹)

(13)

by [9, Proposition 2.7], which satisfies the statement of our lemma. For` >1, the generating function ofd_`(n²)ϕ(n²)is:

ζd_`,ϕ(s) := X

n∈A⁺

d`(n²)ϕ(n²)q^−s^degⁿ

=Y

P

(

(1−q⁻^deg^P)

"

(1 +q^{(2−s) deg}² ^P)^`+ (1−q^{(2−s) deg}² ^P)^` 2(1−q^{(2−s) deg}^P)^`

#

+q⁻^deg^P ) (2.1)

=ζ

`(`+1) 2

A (s−2)·c_` s−2

2

,

which has a pole of order ^`(`+1)₂ at s= 3. Set u=q^−s, ζ˜_d_`_,ϕ(u) := ζ_d_`_,ϕ(s), and c˜_`(u) :=

c_` ^s−2₂

. Then Theorem 2.1 (or [9, Theorem 17.4]) witha= 3 andw= ^`(`+1)₂ gives us the desired result, since the leading coefficient of Laurent series ofζ˜d_`,ϕ(u)atu=q⁻³is equal to

(−q⁻³)^`(`+1)² ·˜c`(q⁻³).

For eachn∈A⁺, we define

ν(n) :=Y

P|n

1−q^{−2 deg}^P⁻¹ . (2.2)

Applying Theorem 2.1 witha= 3andw= ^`(`+1)₂ tod`(n²)ϕ(n²)ν(n), we have Lemma 2.4. Let `be a positive integer, then

X

n∈A+ : degn=N

d`(n²)ϕ(n²)ν(n) =c^∗_`(1/2)·q^3N ·N^`(`+1)² ⁻¹ Γ_`(`+1)

2

+O q^3N·N^`(`+1)² ⁻¹ N

! ,

asN→ ∞.

Proof. The generating function of d`(n²)ϕ(n²)ν(n)is:

ζ_d_`_,ϕ,ν(s) := X

n∈A⁺

d_`(n²)ϕ(n²)ν(n)q^−s^degⁿ

=Y

P

( 1 1 +q⁻^deg^P

""

(1 +q^{(2−s) deg}² ^P)^−`+ (1−q^{(2−s) deg}² ^P)^−`

2

#

+q⁻^deg^P

#) (2.3)

=ζ

`(`+1) 2

A (s−2)·c^∗_` s−2

2

, which has a pole of order ^`(`+1)₂ ats= 3.

Set u = q^−s, ζd_`,ϕ,ν(s) = ˜ζd_`,ϕ,ν(u), and c˜^∗_`(u) := c^∗_` ^s−2₂

. Then Theorem 2.1 (or [9, Theorem 17.4]) with a = 3 and w = ^`(`+1)₂ gives us the desired result, since the leading coefficient of Laurent series ofζ˜d_`,ϕ,ν(u)atu=q⁻³ is equal to

(−q⁻³)^`(`+1)² ·˜c^∗_`(q⁻³).

We finally study the asymptotic formulas of the general divisor functiond_`and the Gauss sumG˜_e. Letg∈A⁺ and

L_g(s, γ_p,q·χ_e) :=Y

P-g

1−γ_p,q^deg^Pχ_e(P)q^−sdeg^P⁻¹

, on<(s)>1

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for each monic polynomiale. We have

Lemma 2.5. Write e=e1e²₂, wheree1∈A⁺ is a square-free polynomial, and e2 is a monic polynomial. In the region<(s)>3/2

X

n∈A+ : (n,g)=1

d`(n) ˜Ge(n)q^−s^degⁿ =Lg(s−1/2, γp,q·χe₁)^`Y

P-g

G_P,d

`,G˜_e(s) :=Lg(s−1/2, γp,q·χe₁)^`G_d

`,G˜_e,g(s), whereG_P,d

`,G˜_e(s)is defined as follows:

G_P,d

`,G˜_e(s) :=

1−γ_p,q^deg^P·χ_e₁(P)q(1/2−s) degP^`





∞

X

t≥0

d_`(P^t) ˜G_e(P^t)q^−st^deg^P



.

ThenG_d

`,G˜_e,g(s)is holomorphic on<(s)>1. Hereγp,q is defined in (1.2).

Proof. The Euler product ofG_d

`,G˜e,g(s)follows from the multiplicativity ofG˜eandd`. By Lemma 1.4, we see that forP -eand`≥3,

G_P,d

`,G˜e(s) =

1−γ_p,q^deg^Pχ_e₁(P)q(1/2−s) degP`

1 +γ^deg_p,q ^P·`·χ_e₁(P)q(1/2−s) degP

=1−`(`+ 1)

2 q^{(1−2s) deg}^P+O(q(3/2−3s) degP

) which implies thatG_d

`,G˜_e,g(s)is holomorphic on<(s)>1. When`= 1or2, we use the same method to prove thatG_d

`,G˜e,g(s)is holomorphic on<(s)>1.

Corollary 2.6. If e∈A andg∈A⁺, then we have, for any δ >0,

X

n∈A+ :degn=N (n,g)=1

d`(n) ˜Ge(n)

q^(1+δ)N, if e6=; q⁽³²^+δ)N, if e=, asN→ ∞.

Proof. LetLg(s, χe) :=Y

P-g

(1−χe(P)q^−s^deg^P)⁻¹ for<(s)>1. SetLg(s, γp,q·χe) = ˜Lg(u, γp,q· χe)andL˜g(u, χe) =Lg(s, χe)withu=q^−s. Then

L˜_g(u, γ_p,q·χ_e) =

L˜g(u, χe), ifγp,q = 1;

L˜_g(−u, χ_e), ifγ_p,q =−1.

Ifeis not square, then the functionL˜_g(u, γ_p,q·χ_e)is holomorphic on C. Ifeis square, then L˜g(u, γp,q·χe)is holomorphic on{u:|u|< q^−3/2}and has a pole at the circle{u:|u|=q^−3/2}.

Thus, for allg ∈A⁺, the function G_d

`,G˜_e,g(s) is holomorphic on <(s) >1 by the above lemma, so Theorem 2.1 and Corollary 2.2 witha= 3/2 for the casee=and a= 1for the casee6=imply that, for anyg∈A⁺,

X

n∈A+ :degn=N (n,g)=1

d`(n) ˜Ge(n)

q^(1+δ)N, ife6=; q⁽³²^+δ)N, ife= for anyδ >0.

Whendegm=M is an even number, we will encounter extra contribution which is

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Lemma 2.7. Let `be a positive integer. Then X

n∈A⁺

d_`(n) =q^N·N^`−1

Γ(`) +O q^NN^`−2

asN → ∞.

This proof is simpler than the above cases, so we omit it.

2.1. Limit distributions. A distribution function is a non-decreasing functionf :R→[0,1]

which is right continuous and satisfiesf(−∞) = 0andf(∞) = 1. In 1931, M. Fréchet’s and J. Shohat’s proved that (cf. [5, Lemma 1.43])

Lemma 2.8. If all the fN(x) from a sequence of distribution functions fN(x) have finite moments αT(N) =

ˆ

R

x^TdfN(x) of every order and if αT(N) → βT as N → ∞ for each T ∈N, then the βT are the moments of some distribution function f(x). If, moreover, f(x) is uniquely determined by its moments, then asN→ ∞the sequencefN(x)converges tof(x) at each point of continuity off(x).

Now to justify the application of the above lemma, we still need one lemma (cf. [5, Lemma 1.44]).

Lemma 2.9. Let α0 = 1,α1, . . . αT, . . .be the moments of some distribution function f(x), each being assumed finite, and suppose that the series

∞

X

T=0

α_T T!τ₀^T

is absolutely convergent for some τ0 > 0. Then f(x) is the unique distribution function with moments α0, α1, α2, . . .. Moreover the characteristic function φf(y) : R → C of the distributionf has the representation

φ_f(y) =

∞

X

T=0

α_T T!(iy)^T for|y|< τ0.

The proof of Corollary 0.3 is similar to Corollary 0.8. We only prove one of them.

Proof of Corollary 0.3.

For a fixeds₀∈Rwiths₀≥1, the real value function f_M(x, s₀) := 1

q^M#{m∈A⁺: degm=M andL(s₀, χ_m)≤x}, x∈R are distribution functions for allM ∈N. Theorem 0.1 says that

1

q^ML(s0, M, `)?=ζA(2s0)^`(`+1)² ·c`(s0) = X

n∈A⁺

d`(n²)·ϕ(n)

|n|^2s⁰⁺¹ =r`(s0), asM → ∞, where?is either S,I orR. According to Lemma 2.3,

r_`(s₀) = X

N≥0





 X

n∈A+ : degn=N

d_`(n²)ϕ(n²)





q^−(2s⁰^+2)N 1

1−q^(2s⁰^−1−δ), for anyδ >0, we obtain that

1 +X

`=1

r`(s0)x^`/`!