Moments of non-normal number fields - II

Suppose that $K$ is a field extension of degree $d$ over $\mathbb{Q}$ (that is, a number field). Let $a_{K}(m)$ (for $m\in\mathbb{N}$) denote the number of integral ideals in $K$ of norm equal to $m$. Let $s$ be a complex number. The Dedekind zeta function of $K$ can be expressed as

It can be shown that $a_{K}(m)$ is a multiplicative function and satisfies the bound

for any positive $\epsilon$. Thus the series (1.1) converges absolutely in the half plane $\Re(s)>1$ where it can be expressed as the following Euler product,

Furthermore $\zeta_{K}(s)$ has a meromorphic continuation to the whole complex plane with a simple pole at $s=1$ and satisfies a functional equation connecting values at $s$ and $1-s$ (see [Neu99, §5, Chapter VII]).

Dedekind zeta functions are natural generalizations of the Riemann zeta function for number fields. As with the Riemann zeta function, the behavior of $\zeta_{K}(s)$ inside the critical strip $0<\Re(s)<1$ is quite mysterious. There are many aspects of the behavior of Dedekind zeta function inside the critical strip that are of interest. In this paper, we focus on understanding the “moments” along the critical line (that is $\Re(s)=\frac{1}{2}$)

Obtaining precise asymptotic for the above integral is a very hard problem, and even the base case of $K=\mathbb{Q}$ poses serious difficulties. Thus we turn to the discrete analogue of the above problem, which maybe more accessible. Namely, we ask if we can estimate

for positive integral values of $l$. The case when $l=1$ is classical and maybe deduced as a consequence of the meromorphic continuation of $\zeta_{K}(s)$. This is also analogous to obtaining estimates for averages of the higher order divisor function (often called the Piltz divisor problem). This problem was first considered by Chandrasekharan and Narasimhan for the case when $K$ was Galois over $\mathbb{Q}$ and when $l=2$ [CN63] and was generalized by Chandrasekharan and Good for arbitrary $l$ [CG83] (but still when $K$ was Galois over $\mathbb{Q}$). A particular non-Galois case was settled by Fomenko [Fom07] and later improved upon by Lü [L1̈3]. By different methods, this was further generalized in a recent work of the author (along with Kalyan Chakraborty) for many families of non-Galois number fields [CK22]. However, the general problem still remained unsolved. The purpose of this paper is to estimate $M_{K}^{l}(T)$ for any number field $K$ and any positive integer $l$ thereby completing the solution to this problem. We also provide a unified treatment which reproduces many of the special cases treated in previous work. The final result conforms to expectations in that the main term is of the order $T$ times a power of $\log(T)$.

Before we state the main theorem, we introduce and fix the following notation throughout the paper. Every representation that we consider will be over $\mathbb{C}$. Let $K$ be as above and $L$ be its Galois closure. The degree of $K$ shall be denoted by $d$. Denote the Galois groups $Gal(L/\mathbb{Q})$ as $G$ and its subgroup $Gal(L/K)$ as $H$. Let $1_{H}$ denote the trivial representation of $H$. Denote the corresponding induction to $G$ as $\rho_{H}$ and its character as $\chi_{\rho_{H}}$. We now have the following theorem.

As we have mentioned before, it is of interest to find asymptotics for $I_{K}^{(l)}(T)$ and often such estimates are tied to the estimates for $M_{K}^{(l)}(T)$. When $K$ is Galois over $\mathbb{Q}$ (equivalently $H$ is the trivial subgroup), lower bounds on $I_{K}^{(l)}(T)$ of the expected order of magnitude are known unconditionally [AF12]. Conditionally on the generalized Riemann hypothesis, upper bounds of the correct order of magnitude (except for an $\epsilon$) are also known [MTB14]. Below, we provide analogous lower bounds for the non-Galois case. It would be convenient to define

The above lower bound is of the expected order of magnitude. To see this we briefly recall some definitions regarding the Selberg class [Sel92] and direct the reader to [Mur94] for more details. Define the class $\mathcal{S}$ to consist of Dirichlet series $F(s):=\sum_{n=1}^{\infty}\frac{a_{F}(n)}{n^{s}}$ which satisfy the following properties:

1. (1)

   (Region of convergence) The series defining $F(s)$ converges absolutely for $\Re(s)>1$.

2. (2)

   (Analytic continuation) $F(s)$ extends to a meromorphic function so that for some integer $m\geqslant 0$, $(s-1)^{m}F(s)$ is an entire function of finite order.

3. (3)

   (Functional equation) There are numbers $Q>0,\alpha_{i}>0,\Re(r_{i})\geqslant 0$ such that

   $$\Phi(s):=Q^{s}\prod_{i=1}^{d}\Gamma(\alpha_{i}s+r_{i})F(s)$$

   satisfies $\Phi(s)=w\overline{\Phi(1-\overline{s})}$ for some complex number $w$ with $|w|=1$.

4. (4)

   (Euler product) $F(s)$ can be written as the product $\prod_{p}F_{p}(s)$ where $F_{p}(s)=\exp\left(\sum_{k=1}^{\infty}b_{p^{k}}/p^{ks}\right)$ where $b_{p^{k}}=\mathcal{O}(p^{k\theta})$ for some $\theta<1/2$.

5. (5)

   (Ramanujan hypothesis) $a_{F}(n)=\mathcal{O}(n^{\epsilon})$ for any fixed $\epsilon>0$.

A function $F\in\mathcal{S}$ is called primitive if $F$ cannot be written as a product of any two elements of $\mathcal{S}$ except for $F=1\cdot F$. Selberg made the following conjectures about the elements in $\mathcal{S}$.

It is expected that for an irreducible representation $\xi$ of the Galois group $G$, the Artin $L$ function $L(s,\xi)$ is a primitive element of the Selberg class. If $\{\xi_{i}\}$ is a complete list of irreducible representations of $G$, then $\zeta_{K}(s)=\prod_{i}L(s,\xi_{i})^{e_{i}}$ is a decomposition of $\zeta_{K}(s)$ into primitive elements (inside the Selberg class), where we have set $e_{i}:=\langle\rho_{H},\xi_{i}\rangle_{G}$. In this situation [Hea21, Conjecture 5] along with Lemmas 4 and 7 leads to the conjecture

for $k>0$ and some constant $c(k,K)$ depending on $k$ and $K$. Finally, if $K$ is Galois over $\mathbb{Q}$, $H$ will be the trivial subgroup and $\beta_{K}=d$. The only case of (1.8) known to be true is when $K$ is a quadratic extension of $\mathbb{Q}$ and when $k=1$ [Mot70].

Regarding upper bounds for the moments, very little is known unconditionally. From their approximate functional equation, Chandrasekharan and Narasimhan ([CN63, see pg. 61]) were able to deduce that

for some constant $c>0$ and $d>2$. As a consequence of Theorem 1, we may improve this as follows.

The fact that this is indeed an improvement follows from Lemma 6.

For the convenience of the reader, we compile some basic facts which we shall use throughout the proofs.

In the remainder of this paper, it is convenient, in many instances, to restrict ourselves to unramified primes. As we are omitting only finitely many primes, this does not affect the nature of our results but for the exact constants. The order of growth shall remain the same. The strategy of the proof is essentially that of [CK22] adopted and generalized with modifications for the current needs. Define

From (1.2), $D_{l}(s)$ is absolutely convergent for $\Re(s)>1$ where the Euler product expression is valid.

We first note two applications of the Chebotarev density theorem.

The proof of the following lemma and corollary follow almost verbatim to the proofs above and hence we omit the details. We use the Chebatorev density theorem in the following form

The proof follows along the same lines as [CN63, §6], where we use Theorem 1 in place of their [CN63, Theorem 3]. The key improvement of the bound on the off-diagonal terms is given in the following lemma. First we observe from Theorem 1 and the results in the previous section, that

Now we may follow [CN63, §6] almost verbatim and complete the proof of Theorem 3. The following lemma applied to $G=Gal(L/\mathbb{Q})$ and $H=Gal(L/K)$ implies that Theorem 3 is indeed an improvement over (1.9).

We first have the following lemma.

As mentioned above, from the general conjectures regarding the primitivity of Artin $L$ functions, $\zeta_{K}(s)=\prod_{i=1}^{n}L(s,\xi_{i})^{e_{i}}$ is a decomposition of $\zeta_{K}$ into primitive elements of $\mathcal{S}$ (where as before $\{\xi_{i}\}$ is a complete set of irreducible representations of $G$). If this is the case, Lemma 7 would follow from Selberg’s Conjecture B (see [Mur94, Proposition 2.5(a)] and [Mur94, Corollary 3.2]). In any case the results of this paper would follow if we assume Selberg’s conjectures. As was shown in [Mur94], Selberg’s conjectures imply the Langlands reciprocity conjecture (or the strong Artin conjecture) from where we may show the required asymptotic in Theorem 1 with a power saving error term. In fact it is expected that the right hand side of (1.6) should be of the form $TP_{l}(\log(T))+\mathcal{O}(T^{1-\theta})$ for some polynomial $P_{l}$ of degree $\mathsf{m}_{l}$ and for some $\theta>0$ (as in the main theorem of [CK22]).

The author wishes to thank Prof. Sudhir Pujahari for his encouragement.