$\Omega$-result for the index of composition of an integral ideal

1.   Introduction

Let $K$ be an algebraic number filed of degree $d$ and $O_K$ be the ring of integers of $K$. For each integral ideal $\mathfrak{A}\in O_K$, $\mathfrak{A} = \mathfrak{P}_1^{e_1}\cdots \mathfrak{P}_g^{e_g}$, where the $\mathfrak{P}_i(i = 1, ..., g)$ are prime ideals of $O_K$, this expression is unique up to the order of the factors. Motivated by ^[17], we define $\lambda(\mathfrak{A}): = \frac{\log N(\mathfrak{A})}{\log\gamma(\mathfrak{A})}$ be the index of composition of $\mathfrak{A}$, where $N(\mathfrak{A})$ is the norm of ideal $\mathfrak{A}$ and $\gamma(\mathfrak{A}) = \prod_{\mathfrak{P}|\mathfrak{A}}N(\mathfrak{P})$. We write $\lambda(\mathfrak{A}) = \gamma(\mathfrak{A}) = 1$ if $\mathfrak{A} = O_K$. The index of composition of an integral ideal measures the multiplicity of its prime factors.

Before stating our main results, we introduce some notations. The Dedekind zeta-function for the field $K$ is defined by

where $a_n$ is the number of integral ideals of $K$ with norm $n$. $\zeta_K(s)$ can be analytically continued to the whole complex plane, $s = 1$ is a simple pole with residue

where $r_1, r_2$ denote the number of real and complex places respectively, $R_K$ is the regular of $K$, $d(K)$ is the discriminant of $K$ and $\omega_K$ is the order of the group of units. We know that $a_n\leq(\tau(n))^d$, where $\tau(n)$ is the number of divisors of $n$. Let

Then (see ^[5,8,9,11])

where

In ^[15], Zhang and Zhai obtained a series of results about the mean value of $\lambda^{\pm k}(\mathfrak{A})$. The results imply that the average order of $\lambda(\mathfrak{A})$ is $\rho_K$. Also they found that the mean value of $\lambda^{-1}(\mathfrak{A})$ has a close connection with the zero free region of $\zeta_K(s)$ and got that

where

If $K$ is a quadratic or cubic number field, Zhang and Zhai ^[16] proved the asymptotic formula

where

and $C_{1}, C_{2}$ are computable constants, $c > 0$ is a positive constant. Assuming the Riemann Hypothesis for $\zeta_K(s)$ is true, Zhang and Zhai ^[16] used the estimation of exponential sum and convolution method to get

It is natural to consider how well the main term of $\sum_{N(\mathfrak{A})\leq x}\lambda^{-1}(\mathfrak{A})$ approximates it, that is, what can be said about $\Omega$- result (for example, see ^[6,7,18]). In this paper, we shall get the following result.

Theorem 1.1. Let $K$ be a quadratic or cubic number field. Then we have

where $C_{i} (i = 1, 2, 3)$ are computable constants and the error term $R_0(x)$ satisfies

Remark. In order to prove Theorem 1.1, we will follow the line in Pintz ^[12] and Nowak ^[10], using Mellin transform and constructing an auxiliary function $g(s)$ with some properties. Because $\lambda(\mathfrak{A})$ is not multiplicative, it is not easy to get the generating series of $\lambda^{-1}(\mathfrak{A})$. Instead we shall study the mean value of $\sum_{N(\mathfrak{A})\leq x}\log\gamma(\mathfrak{A})$. In fact, as a function of $z$, $\gamma^{z}(\mathfrak{A})$ is regular for $|z|\leq\varepsilon$, then we can differentiate it and set $z = 0$ to get the mean value and generating series for $\log\gamma(\mathfrak{A})$. Theorem 1.1 can follows from the lower bound for the error term of $\sum_{N(\mathfrak{A})\leq x}\log\gamma(\mathfrak{A})$.

It is easy to see that (1.3) implies the following $\Omega$-result.

Theorem 1.2. Let $K$ be a quadratic or cubic number field. Then we have

Notation. Throughout the paper $\varepsilon$ always denotes a fixed but sufficiently small positive constant. We write $f(x)\ll g(x)$, or $f(x) = O(g(x))$, to mean that $|f(x)|\leq C g(x)$. $\sum_{n\sim N}$ denote that the sum over $N < n\leq 2N.$ $f(x) = \Omega(g(x))$ means that there exists a suitable constant $C > 0$ such that $|f(x)| > C g(x)$ holds for a sequence $x = x_n$ such that $\lim\limits_{n\rightarrow\infty}x_n = \infty.$

2.   The mean value of $\log\gamma(\mathfrak{A})$

In this section, suppose $\varepsilon > 0$ is a small positive constant, $z$ is a complex number such that $|z|\leq\varepsilon$. Let $s = \sigma+it$ be a complex number with $\Re(s-z) > 1$. Define

Lemma 2.1. For $|z\leq\varepsilon$, we have

where $G_1(s, z)$ can be expanded into a Dirichlet series of $s$, which is absolutely convergent for $\sigma > \frac{1}{5}+\varepsilon$.

Proof. By Euler product representation, we have

where $\zeta_K(s)$ is the Dedekind $\zeta$-function, and

where

Arguing similarly, we can get

where

By the similar method as before, we know that $G_1(s, z)$ can be written as the product of Dedekind $\zeta$-functions. If we note that $|z|\leq\varepsilon$, then $G_1(s, z)$can be expanded to a Dirichlet series, which is absolutely convergent for $\Re s > 1/5+\varepsilon$.

To obtain the mean value of $\sum_{N(\mathfrak{A})\leq x}\log\gamma(\mathfrak{A})$, we need the following Lemma.

Lemma 2.2. If $\sigma > 1$, then

Let $k\geq 0$ be an integer. Uniformly for $\frac{1}{2}\leq\sigma\leq1$, we have

where $\zeta_K^{(k)}(s)$ is the $k$-th derivative of $\zeta_K(s)$.

Proof. The order of $\zeta_K(s)$ can be found in [16,Lemma 2.3]. For the cases $k\geq 1$ of (2.2), we use the Cauchy derivative formula to get

Let $z = s+R e^{i\theta}$ and $R = 1/\log(|t|+2)$. Then the above formula can be written as

Therefore Lemma 2.2 follows from the order of $\zeta_K(s)$ $(k = 0$ in (2.2)).

Proposition 2.3. We have

where $c_i\ (1\leq i\leq3)$ are computable constants, and for $Y\rightarrow\infty$, the error term $E_0(x)$ satisfies

Proof. By Lemma 2.1, we have

Firstly, we take the first partial derivative with respect to $z$ form both sides of (2.3), and put $z = 0$ to get

where

and $G_1(s, 0)$ is absolutely convergent for $\Re s > \frac{1}{5}+\varepsilon$. We can easily see that $\sum_\mathfrak{A}\frac{\log\gamma(\mathfrak{A})}{N^s(\mathfrak{A})}$ has a pole of order $2$ at $s = 1$ and poles of order $1$ at $s = \frac{1}{2}, \frac{1}{3}$, which prompts us to consider that $\sum_{N(\mathfrak{A})\leq x}\log\gamma(\mathfrak{A})$ should have the following asymptotic formula

where $E_0(x) = O(x^{\frac{1}{4}+\varepsilon})$.

Following the idea of Pintz ^[12] and Nowak ^[10], we use the Mellin transform to get

for $\Re s > 1$. Now we deal with the first term of (2.4). By the partial integration and (2.4)., we have

Let $K(s) = \zeta_K(s)\zeta_K(4s)G_1(s, 0)(\zeta_K^{'}(2s)\zeta_K(3s)+\zeta(2s)\zeta_K^{'}(3s))$. After computing another four terms of (2.4), we can get

where

here $M(s) = (2s-1)(3s-1)\left(\rho_K+c_1 (s-1)\right)+(s-1)^2\left(2c_2(3s-1)+3c_3 (2s-1)\right)$. It is easy to see that $F(s)$ is an entire function for $\Re s > \frac{1}{5}+\epsilon$. We choose $z_0 = \frac{1}{4}+i\beta_0$, according to the results of ^[1] (or ^{[2,3,4,13,14]}), we get $2z_0$ is a single zero of the Dedekind $\zeta$-function and $\zeta_K(z_0)\zeta_K(3z_0)G_1(z_0, 0)\neq0$. In addition,

We write

which is regular in $\Re s > -2$, and

is regular in $\Re s > \frac{1}{5}+\epsilon$ apart from a simple pole at $s = z_0$, since

Using the order of $\zeta_K(s)$ and $\zeta_K'(s)$ (Lemma 2.2), we know that the integrals

converge for $\beta\in\{\frac{1}{5}, 2\}$ as $|t|\rightarrow\infty$. Now, for $\eta > 0$, we define a weight function

which satisfies

Therefore,

For $Y$ large, we shift the line of integration to $\Re s = \frac{1}{5}$, then we have

where

By (2.6), we can evident that

as $Y\rightarrow\infty$. On the other hand, by (2.5), we can obtain

Consequently, for $Y\rightarrow\infty$, we have

3.   Proof of Theorem 1.1

Now we prove Theorem 1.1. From Proposition 2.3, we get

with

where $Y\rightarrow\infty$. Using partial integration, we get

where

Taking the derivative of the above formula we get $E_0'(x) = R_0'(x)\log x$. Integrating both sides with respect to $x$, we have

And also

We prove Theorem 1.1 by contradiction. Suppose that

where $\epsilon$ is a small constant. Thus we have

We use dyadic arguments to the inner integral to get

By (3.5), we can obtain

Inserting (3.8) into (3.6), we have

which contradict with Proposition 2.3. Then we have

4.   Conclusions

For each integral ideal $\mathfrak{A}$, it is essential to measure the multiplicity of its prime ideal factors. In this paper, we define $\lambda(\mathfrak{A}): = \frac{\log N(\mathfrak{A})}{\log\gamma(\mathfrak{A})}$ to be the index of composition of $\mathfrak{A}$ and consider how well the main term of $\sum_{N(\mathfrak{A})\leq x}\lambda^{-1}(\mathfrak{A})$ approximates it, that is, what can be said about $\Omega$-results for the index of composition of integral ideal. The results imply that the average order of $\lambda(\mathfrak{A})$ is $\rho_K$.

Acknowledgments

The authors would like to express their gratitude to the referee for his or her careful reading and valuable suggestions. This work was supported by National Natural Science Foundation of China (Grant Nos. 11771256, 11971476).

Conflict of interest

The authors declare there is no conflict of interest.