The value-distribution of artin L-functions associated with cubic fields in conductor aspect

Abstract

Arising from the factorizations of the Dedekind zeta-functions of cubic fields, we obtain Artin L-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin L-functions for families of non-Galois cubic fields in conductor aspect. For this, we apply asymptotic formulas for the counting functions of cubic fields proved by Bhargava et al. (Invent Math 193 (2):439-499, 2013) or Taniguchi and Thorne (Duke Math J 162(13): 2451-2508, 2013). Then the Artin L-functions associated with cubic fields are connected with random variables called the random Euler products. We construct a density function for the random Euler product, whose Fourier–Laplace transform has an infinite product representation. Furthermore, we prove that various mean values of the Artin L-functions are represented by integrals involving this density function. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields. We show a formula for the sum involving the class numbers which is regarded as a cubic analogue of Gauss’ formula on quadratic class numbers.

Appendix: Results on the logarithmic derivative

In the above sections, we proved several results on values $\log {L}(\sigma ,\rho _K)$. We obtain similar results for $(L'/L)(\sigma ,\rho _K)$, which are listed in this section. Note that the value $(L'/L)(1,\rho _K)$ is connected with the Euler–Kronecker constant $\gamma _K$ defined by

$$\begin{aligned} \gamma _K =\lim _{s \rightarrow 1} \left( \frac{\zeta '_K}{\zeta _K}(s)+\frac{1}{s-1}\right) . \end{aligned}$$

By definition, $\gamma _{\mathbb {Q}}$ is equal to Euler’s constant $\gamma =0.577\ldots$. Thus we obtain

$$\begin{aligned} \frac{L'}{L}(1,\rho _K) =\gamma _K-\gamma \end{aligned}$$

for any non-Galois cubic field K since $L(s,\rho _K)=\zeta _K(s)/\zeta (s)$. The proofs for the results in this section are omitted unless we need special remarks arising from the difference between $\log {L}(\sigma ,\rho _K)$ and $(L'/L)(\sigma ,\rho _K)$. Let ${\mathbb {X}}=({\mathbb {X}}_p)_p$ and ${\mathbb {Y}}=({\mathbb {Y}}_p)_p$ be the sequences of independent random elements on $\{A_{\mathfrak {a}} \mid {\mathfrak {a}} \in {\mathscr {A}}\}$ as in Sect. 2. Then we define the random Euler products $L(s,{\mathbb {X}})$ and $L(s,{\mathbb {Y}})$ as in (2.7).

Theorem A.1

For $\sigma >1/2$, there exists a non-negative $C^\infty$-function ${\mathscr {C}}_\sigma$ such that

$$\begin{aligned} {\mathbb {P}}\left( \frac{L'}{L}(\sigma ,{\mathbb {X}}) \in A \right) =\int _{A} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

holds for all $A \in {\mathscr {B}}({\mathbb {R}})$. Furthermore, it satisfies the following properties:

• (i) If $1/2<\sigma \le 1$, we have ${{\,\textrm{supp}\,}}{\mathscr {C}}_\sigma ={\mathbb {R}}$, that is, ${\mathscr {C}}_\sigma (x)$ is not identically zero in any interval on ${\mathbb {R}}$;

• (ii) If $\sigma >1$, the function ${\mathscr {C}}_\sigma$ is compactly supported;

• (iii) Let $\sigma >1/2$. Then the integral

  $$\begin{aligned} \int _{-\infty }^{\infty } e^{ax} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

  is finite for any $a>0$.

Similarly, for $\sigma >2/3$, there exists a non-negative $C^\infty$-function $K_\sigma$ such that

$$\begin{aligned} {\mathbb {P}}\left( \frac{L'}{L}(\sigma ,{\mathbb {Y}}) \in A \right) =\int _{A} {\mathscr {K}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

holds for all $A \in {\mathscr {B}}({\mathbb {R}})$. Furthermore, it satisfies the following properties:

• (i’) If $2/3<\sigma \le 1$, we have ${{\,\textrm{supp}\,}}{\mathscr {K}}_\sigma ={\mathbb {R}}$, that is, ${\mathscr {K}}_\sigma (x)$ is not identically zero in any interval on ${\mathbb {R}}$;

• (ii’) If $\sigma >1$, the function ${\mathscr {K}}_\sigma$ is compactly supported;

• (iii’) Let $\sigma >2/3$. Then the integral

  $$\begin{aligned} \int _{-\infty }^{\infty } e^{ax} {\mathscr {K}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

  is finite for any $a>0$.

Denote by ${\mathscr {E}}_\sigma (X)$ the subset of $L_3^\pm (X)$ such that ${\mathscr {E}}_\sigma (X)=E(X)$ for $\sigma _1<\sigma \le 1$ and ${\mathscr {E}}_\sigma (X)=\emptyset$ for $\sigma >1$, where $\sigma _1$ is a real number for which (2.10) holds, and E(X) is defined by (2.11). Then we define ${\mathscr {M}}_{z, \sigma }^{\pm }(X)$ as

$$\begin{aligned} {\mathscr {M}}_{z, \sigma }^{\pm }(X) =\sum _{K \in L_3^\pm (X) \setminus {\mathscr {E}}_\sigma (X)} \exp \left( z \frac{L'}{L}(\sigma ,\rho _K)\right) \end{aligned}$$

as an analogue of the z-th moment $M_{z, \sigma }^{\pm }(X)$ given by (1.9).

Theorem A.2

Let $\sigma _1$ be a real number for which (2.10) holds. Then there exists an absolute constant $\delta >0$ such that

$$\begin{aligned} {\mathscr {M}}_{z, \sigma }^{\pm }(X) =C^\pm \frac{1}{12\zeta (3)} X \int _{-\infty }^{\infty } e^{zx} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} +O\left( X\exp \left( -\delta \frac{\log {X}}{\log \log {X}}\right) \right) \end{aligned}$$

(A.1)

holds for $\sigma >\sigma _1$ with $z \in {\mathbb {C}}$ satisfying $|z| \le b_\sigma R_{\sigma }(X)$, where $b_\sigma$ is a positive constant, and $R_{\sigma }(X)$ is defined as in (2.13). The implied constant in (A.1) depends only on $\sigma$.

Note that the subset ${\mathscr {E}}_\sigma (X)$ differs from $E_\sigma (X)$ at $\sigma =1$. We hereby explain the reason why this difference arises. In the same line as (5.15), one can show the asymptotic formula

$$\begin{aligned}&\sum _{K \in L_3^\pm (X) \setminus E(X)} \exp \left( z \frac{L'}{L}(\sigma ,\rho _K)\right) \\&\quad =C^\pm \frac{1}{12\zeta (3)} X \int _{-\infty }^{\infty } e^{zx} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} +O\left( X\exp \left( -\delta \frac{\log {X}}{\log \log {X}}\right) \right) \end{aligned}$$

for $\sigma >\sigma _1$ and $|z| \le b_\sigma R_\sigma (X)$, where $\delta$ is an absolute positive constant. To prove formula (A.1) at $\sigma =1$ with ${\mathscr {E}}_1(X)=\emptyset$, one needs to obtain the upper bound

$$\begin{aligned} \sum _{K \in E(X)} \exp \left( z \frac{L'}{L}(\sigma ,\rho _K)\right) \ll X\exp \left( -\delta \frac{\log {X}}{\log \log {X}}\right) \end{aligned}$$

(A.2)

for $\sigma =1$ and $|z| \le b_\sigma R_\sigma (X)$. This is true for $\sigma >1$ due to $(L'/L)(\sigma ,\rho _K) \ll 1$ and $\# E(X) \ll X^{1-\delta }$. However, we know just

$$\begin{aligned} \frac{L'}{L}(1,\rho _K) \ll \log {X} \end{aligned}$$

for $K \in L_3^\pm (X)$ without any assumptions, which prevents us from obtaining (A.2) at $\sigma =1$. A related difficulty on studying the distribution of values $(L'/L)(1,\chi _d)$ was discussed in the Ph.D. thesis of Mourtada [43]. See also Lamzouri [35] for more information about the Euler–Kronecker constants of quadratic fields.

Theorem A.3

Assume GRH and upper bound (2.14) for each $\epsilon >0$ with some constants $\alpha$ and $\beta$ such that $0<\alpha <5/6$. Let $\sigma _2$ be as in (2.15). Then there exists a constant $\delta =\delta (\sigma )>0$ such that

$$\begin{aligned} {\mathscr {M}}_{z, \sigma }^{\pm }(X)&=C^\pm \frac{1}{12\zeta (3)} X \int _{-\infty }^{\infty } e^{zx} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \nonumber \\&\quad +K^\pm \frac{4\zeta (1/3)}{5\varGamma (2/3)^3\zeta (5/3)} X^{5/6} \int _{-\infty }^{\infty } e^{zx} {\mathscr {K}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} +O\left( X^{5/6-\delta }\right) \end{aligned}$$

(A.3)

holds for any $\sigma >\max (\sigma _2,2/3)$ with $z \in {\mathbb {C}}$ satisfying $|z| \le {\tilde{b}}_\sigma {\tilde{R}}_{\sigma }(X)$, where ${\tilde{b}}_\sigma$ is a positive constant, and ${\widetilde{R}}_\sigma (X)$ is defined as in (2.17). The implied constant in (A.3) depends only on $\sigma$.

Corollary A.2

There exists an absolute constant $\delta >0$ such that

$$\begin{aligned} \sum _{K \in L_3^\pm (X) \setminus E(X)} \exp (z \cdot \gamma _K)&=C^\pm \frac{1}{12\zeta (3)} X e^{\gamma z} \int _{-\infty }^{\infty } e^{rx} {\mathscr {C}}_1(x) \,\frac{dx}{\sqrt{2\pi }} \\&\quad +O\left( X \exp \left( -\delta \frac{\log {X}}{\log \log {X}}\right) \right) \end{aligned}$$

holds for any $z \in {\mathbb {C}}$ with $|z| \le b_1 R_1(X)$, where the implied constant is absolute.

Corollary A.3

Assume GRH and upper bound (2.14) with some constants $\alpha$ and $\beta$ such that $3\alpha +\beta <5/2$. Then there exists an absolute constant $\delta >0$ such that

$$\begin{aligned} \sum _{K \in L_3^\pm (X)} \exp (z \cdot \gamma _K)&=C^\pm \frac{1}{12\zeta (3)} X e^{\gamma z} \int _{-\infty }^{\infty } e^{zx} {\mathscr {C}}_1(x) \,\frac{dx}{\sqrt{2\pi }} \\&\quad +K^\pm \frac{4\zeta (1/3)}{5\varGamma (2/3)^3\zeta (5/3)} X^{5/6} e^{\gamma z} \int _{-\infty }^{\infty } e^{zx} {\mathscr {K}}_1(x) \,\frac{dx}{\sqrt{2\pi }} \\&\quad +O\left( X \exp \left( -\delta \frac{\log {X}}{\log \log {X}}\right) \right) \end{aligned}$$

holds for any $z \in {\mathbb {C}}$ with $|z| \le {\tilde{b}}_1 {\tilde{R}}_1(X)$, where the implied constant is absolute.

Define the quantity ${\mathscr {D}}_\sigma ^\pm (X; a)$ as

$$\begin{aligned} {\mathscr {D}}_\sigma ^\pm (X; a) =\frac{\# \left\{ K \in L_3^\pm (X) \,\Big |\, \frac{L'}{L}(\sigma ,\rho _K) \le a \right\} }{N_3^\pm (X)} -\int _{-\infty }^{a} {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

for $\sigma >1/2$ and $a \in {\mathbb {R}}$.

Theorem A.4

Let $\sigma _1$ be a real number for which (2.10) holds. Then we obtain

$$\begin{aligned} \sup _{a \in {\mathbb {R}}} \left| {\mathscr {D}}_\sigma ^\pm (X; a)\right| \ll \frac{1}{R_\sigma (X)} \end{aligned}$$

for $\sigma >\sigma _1$, where $R_\sigma (X)$ is as in (2.13).

Finally, we present a result similar to Theorem 2.5. There exists a difference from the result for $\log {L}(\sigma ,\rho _K)$ when we consider the case $\sigma =1$ by the same reason as in Theorem A.3. For this, we define a subclass of C(S) as

$$\begin{aligned} C^{{{\,\textrm{poly}\,}}}({\mathbb {R}}) =\left\{ \varPhi \in {C}({\mathbb {R}}) \,|\, \varPhi (x) \ll |x|^a\text { with some }a>0 \right\} . \end{aligned}$$

Theorem A.5

Let $\sigma _1$ be a real number for which (2.10) holds. Then the limit formula

$$\begin{aligned} \lim _{X \rightarrow \infty } \frac{1}{N_3^\pm (X)} \mathop {{\mathop {\sum }\nolimits '}}\limits _{K \in L_3^\pm (X)} \varPhi \left( \frac{L'}{L}(\sigma ,\rho _K)\right) =\int _{-\infty }^{\infty } \varPhi (u) {\mathscr {C}}_\sigma (x) \,\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

holds in the following cases:

• $\sigma >1$ and $\varPhi \in C({\mathbb {R}}) \cup I({\mathbb {R}})$;

• $\sigma =1$ and $\varPhi \in C^{{{\,\textrm{poly}\,}}}({\mathbb {R}}) \cup I({\mathbb {R}})$;

• $\sigma _1<\sigma <1$ and $\varPhi \in C_b({\mathbb {R}}) \cup I({\mathbb {R}})$ without assuming GRH;

• $\sigma _1<\sigma \le 1$ and $\varPhi \in C^{\exp }({\mathbb {R}}) \cup I({\mathbb {R}})$ if we assume GRH.