TRANSCENDENCE OF GENERALISED EULER–KRONECKER CONSTANTS

Abstract

We introduce some generalisations of the Euler–Kronecker constant of a number field and study their arithmetic nature.

1 Introduction and preliminaries

In 1740, Euler [2] introduced the Euler–Mascheroni constant, which is defined as

(1.1) $$ \begin{align} \gamma = \lim_{x \to \infty} \bigg( \sum_{n \le x } \frac{1}{n} - \log x \bigg). \end{align} $$

This constant has been extensively studied (see [4]), but many questions about its behaviour are unanswered. For example, it is not known if $\gamma $ is rational or irrational. Diamond and Ford [1] introduced a generalisation of Euler’s constant as follows. For a nonempty finite set of distinct primes $\Omega $ , let $P_{\Omega }$ denote the product of the elements of $\Omega $ and $\delta _{\Omega }= \prod _{p \in \Omega } ( 1- {1}/{p})$ . Then the generalised Euler constant is defined as

$$ \begin{align*} \gamma(\Omega) = \lim_{ x \to \infty} \bigg(\sum_{\substack{n \leq x\\ (n, P_{\Omega})=1}} \frac{1}{n} - \delta_{\Omega} \log x \bigg). \end{align*} $$

Note that when $\Omega = \emptyset $ , we have $P_{\Omega }=1=\delta _{\Omega }$ and $\gamma (\Omega ) = \gamma $ . In this context, Murty and Zaytseva proved the following theorem.

Theorem 1.1 (Murty and Zaytseva, [8]).

At most one number in the infinite list $ \{\gamma (\Omega )\}$ , as $\Omega $ varies over all finite subsets of distinct primes, is algebraic.

We note that $\gamma $ appears as the constant term in the Laurent series expansion of $\zeta (s)$ around $s=1$ . This observation led Ihara [3] to define the Euler–Kronecker constant associated to a number field as follows.

Let ${\mathbf K}$ be a number field of degree n and let ${{\mathcal {O}}}_{\mathbf K}$ denote its ring of integers. The Dedekind zeta function of ${\mathbf K}$ is given by

$$ \begin{align*} \zeta_{{\mathbf K}}(s) = \sum_{(0)\neq {\mathfrak{a}} \subseteq{{{\mathcal{O}}}_{{\mathbf K}}}} \frac{1}{{\mathfrak N}({\mathfrak{a}})^s}, \quad \Re(s)> 1. \end{align*} $$

It has a meromorphic continuation to the entire complex plane with only a simple pole at the point $s=1$ . Its Laurent series expansion around $s=1$ is given by

$$ \begin{align*} \zeta_{\mathbf K}(s) = \frac{\rho_{\mathbf K}}{s -1} + c_{\mathbf K} + \mathrm{O}(s-1), \end{align*} $$

where $\rho _{\mathbf K} \ne 0$ is the residue of $\zeta _{\mathbf K}$ at $s=1.$ Ihara defined the ratio

$$ \begin{align*} \gamma_{\mathbf K} := c_{\mathbf K}/\rho_{\mathbf K} \end{align*} $$

as the Euler–Kronecker constant of ${\mathbf K}$ . In the next section, an expression analogous to (1.1) is given for $\gamma _{\mathbf K}$ .

The aim of this article is to study the arithmetic nature of generalisations of Euler–Kronecker constants. To do so, we introduce some notation. Let $\mathcal {P}_{{\mathbf K}}$ denote the set of nonzero prime ideals $\mathfrak {p}$ of ${{\mathcal {O}}}_{{\mathbf K}}$ and let $\Omega $ be a nonempty subset of $\mathcal {P}_{\mathbf K}$ (possibly infinite) such that

(1.2) $$ \begin{align} \sum_{\mathfrak{p}\in \Omega} \frac {\log {\mathfrak N}(\mathfrak{p})}{{\mathfrak N}(\mathfrak{p})- 1} < \infty. \end{align} $$

For ${\mathbf K} = \mathbb {Q}$ , the set of Pjateckii–Šapiro primes is an example of such an infinite subset. Let $N_{\Omega }=\{ \mathfrak {p}\cap {\mathbb Z} \mid \mathfrak {p} \in \Omega \}$ . We set

$$ \begin{align*} P(\Omega(x)) = \prod_{\mathfrak{p} \in \Omega(x)} \mathfrak{p} \quad \text{and} \quad \delta_{\mathbf K}(\Omega(x)) = \prod_{\mathfrak{p} \in \Omega(x)} \bigg( 1- \frac{1}{{\mathfrak N}(\mathfrak{p})}\bigg), \end{align*} $$

where $\Omega (x) =\{\mathfrak {p} \in \Omega ~| ~ {\mathfrak N}(\mathfrak {p})\leq x\}.$ Then by (1.2), $ \lim _{x \to \infty }\delta _{\mathbf K}(\Omega (x))$ exists and equals

$$ \begin{align*} \delta_{\mathbf K}(\Omega)=\prod_{\mathfrak{p}\in \Omega}\bigg(1-\frac{1}{{\mathfrak N}(\mathfrak{p})}\bigg). \end{align*} $$

Note that $\delta _{\mathbf K}(\Omega )=1$ for $\Omega = \emptyset $ . The generalised Euler–Kronecker constant associated to $\Omega $ is denoted by $\gamma _{\mathbf K}(\Omega )$ and is defined as

$$ \begin{align*} \lim_{ x \to \infty} \bigg( \frac{1}{\rho_{\mathbf K}} \sum_{\substack{0\ne {\mathfrak{a}} \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}({\mathfrak{a}}) \leq x \\ ({\mathfrak{a}}, P(\Omega(x))=1}} \frac{1}{{\mathfrak N}(\mathfrak{a})} - \delta_{\mathbf K}(\Omega(x)) \log x\bigg). \end{align*} $$

In Section 3, we will show that this limit exists. We note that $\gamma _{\mathbf K}(\Omega ) = \gamma _{\mathbf K}$ when $\Omega =\emptyset $ . With this set up, we have the following theorem.

Theorem 1.2. Let $\{\Omega _i\}_{i\in I}$ be a family of subsets of $\mathcal {P}_{\mathbf K}$ satisfying (1.2). Further, suppose that $ N_{\Omega _i} \backslash N_{\Omega _j}$ is nonempty and finite for all $ i,j \in I$ and $i \ne j$ . Then at most one number from the infinite list

$$ \begin{align*} \bigg\{ \frac{\gamma_{\mathbf K}(\Omega_i)}{ \delta_{\mathbf K}(\Omega_i) } ~ \Big | ~ i \in I \bigg\} \end{align*} $$

is algebraic.

We digress here a little to make an interesting observation. For $K={\mathbb Q}$ , it is known by Merten’s theorem that as $x \to \infty $ ,

$$ \begin{align*} \delta_{\mathbb Q}(\Omega(x))\sim \frac{e^{-\gamma}}{\log{x}}. \end{align*} $$

This makes one wonder if $\gamma _{\mathbf K}$ appears as an exponent in the expression for ${\mathbf K}\neq {\mathbb Q}$ . A result of Rosen [9] shows that this is not true in general. More precisely, he showed that as $x \to \infty $ ,

$$ \begin{align*} \delta_{\mathbf K}(\Omega(x))\sim \frac{e^{-\gamma}}{\rho_{\mathbf K} \log{x}}. \end{align*} $$

2 Preliminaries and lemmas

Let ${\mathbf K}$ be a number field of degree n. Throughout this section, ${\mathfrak {p}}$ denotes a nonzero prime ideal of ${{\mathcal {O}}}_{\mathbf K}$ . We recall the following result on counting the number of integral ideals of ${{\mathcal {O}}}_{{\mathbf K}}$ .

Lemma 2.1 [7, Ch. 11].

Let $a_m$ be the number of integral ideals of $\mathcal {O}_{\mathbf K}$ with norm m. Then, as x tends to infinity,

$$ \begin{align*} \sum_{m=1}^{x} a_m = \rho_{{\mathbf K}} x + \mathrm{O}(x^{1-1/n}). \end{align*} $$

Using this result, we find the following expression for $\gamma _{\mathbf K}$ , analogous to (1.1).

Lemma 2.2. For any number field ${\mathbf K}$ , the limit

$$ \begin{align*} \lim_{x \rightarrow \infty} \bigg(\frac{1}{\rho_{\mathbf K}}\sum_{\substack{0\ne {\mathfrak{a}} \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}({\mathfrak{a}}) \leq x}} \frac{1}{{\mathfrak N}({\mathfrak{a}})} - \log x \bigg) \end{align*} $$

exists and equals $\gamma _{{\mathbf K}}$ .

Proof. Applying partial summation and Lemma 2.1, the result follows.

The Möbius function $\mu _{\mathbf K}$ and the von Mangoldt function $\Lambda _{\mathbf K}$ are defined on ${{\mathcal {O}}}_{\mathbf K}$ as follows:

$$ \begin{align*} \mu_{\mathbf K}({\mathfrak{a}}) &= \begin{cases} 1 & \mbox{ if }~ {\mathfrak{a}}={{\mathcal{O}}}_{\mathbf K},\\ (-1)^r & \mbox{ if } {\mathfrak{a}} \mbox{ is a product of } r \mbox{ distinct prime ideals}, \\ 0 & \mbox{ otherwise}; \end{cases} \\ \Lambda_{\mathbf K}({\mathfrak{a}}) &= \begin{cases}\log{\mathfrak N}({\mathfrak{p}}) & \mbox{ if}~ {\mathfrak{a}}={\mathfrak{p}}^m \mbox{ for some } {\mathfrak{p}} \mbox{ and some integer } m \geq 1,\\ 0 & \mbox{ otherwise}. \end{cases} \end{align*} $$

We record the following identities satisfied by these functions which can be derived using techniques similar to [6, Exercises 1.1.2, 1.1.4, 1.1.6].

$$ \begin{align*} \sum_{J \mid I} \frac{\mu_{\mathbf K}(J)}{{\mathfrak N}(J)} & = \prod_{\mathfrak{p}\mid I} \bigg( 1- \frac{1}{{\mathfrak N}(\mathfrak{p})}\bigg),\nonumber\\ \mu_{\mathbf K}(I) \log {\mathfrak N}(I) & = -\sum_{J\mid I} \Lambda_{\mathbf K}(J) \mu_{\mathbf K}(IJ^{-1}). \end{align*} $$

We end this section by stating the key ingredient in the proof of Theorem 1.2.

Lemma 2.3 (Lindemann, [5]).

If $\alpha \neq 0, 1$ is an algebraic number, then $\log {\alpha }$ is transcendental, where $\log $ denotes any branch of the logarithm.

3 Generalised Euler–Kronecker constants

Let $\mathcal {P}_{\mathbf K}$ denote the set of nonzero prime ideals of ${{\mathcal {O}}}_{{\mathbf K}}$ . For any nonempty finite set $\Omega _f \subset \mathcal {P}_{\mathbf K}$ , we set

$$ \begin{align*} P(\Omega_f)= \prod_{\mathfrak{p} \in \Omega_f} \mathfrak{p} \quad \mbox{and} \quad \delta_{\mathbf K}(\Omega_f)=\prod_{\mathfrak{p}\in \Omega_f}\bigg(1-\frac{1}{{\mathfrak N}(\mathfrak{p})} \bigg), \end{align*} $$

with the convention that $P(\Omega _f)=1=\delta _{\mathbf K}(\Omega _f)$ , when $\Omega _f=\emptyset $ . Since $\mathcal {O}_{\mathbf K}$ is a Dedekind domain, every integral ideal can be uniquely expressed as a product of prime ideals. For ideals

$$ \begin{align*} {\mathfrak{a}} = \prod_{{\mathfrak{p}}\in \mathcal{P}_{\mathbf K}} \mathfrak{p}^{v_{ \mathfrak{p}}({\mathfrak{a}})}, \quad {\mathfrak{b}} = \prod_{{\mathfrak{p}}\in \mathcal{P}_{\mathbf K}} \mathfrak{p}^{v_{ \mathfrak{p}}({\mathfrak{b}})}, \end{align*} $$

where all but finitely many $v_{\mathfrak {p}}({\mathfrak {a}}), v_{\mathfrak {p}}({\mathfrak {b}})$ are zero, we define the greatest common divisor (gcd) of ${\mathfrak {a}}$ and ${\mathfrak {b}}$ by

$$ \begin{align*} ({\mathfrak{a}},{\mathfrak{b}})=\gcd({\mathfrak{a}},{\mathfrak{b}})=\prod_{{\mathfrak{p}}\in \mathcal{P}_{\mathbf K}}\mathfrak{p}^{\min(v_{\mathfrak{p}}({\mathfrak{a}}), v_{ \mathfrak{p}}({\mathfrak{b}}))}, \end{align*} $$

where we have denoted $\mathfrak {p}^{0}$ by $\mathcal {O}_{\mathbf K}$ . Hence, if the prime factors of ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are all distinct, $ ({\mathfrak {a}},{\mathfrak {b}}) = \mathcal {O}_{\mathbf K}. $ We notice that $ ({\mathfrak {a}},{\mathfrak {b}})={\mathfrak {a}}+{\mathfrak {b}}$ as $v_{\mathfrak {p}}({\mathfrak {a}}+{\mathfrak {b}})= \min (v_{\mathfrak {p}}({\mathfrak {a}}), v_{\mathfrak {p}}({\mathfrak {b}})).$ From now on, $ \mathcal {O}_{\mathbf K}$ will be denoted by 1.

Lemma 3.1. For a number field ${\mathbf K}$ and a finite set $\Omega _f$ , the limit

$$ \begin{align*} \lim_{ x \to \infty} \bigg( \frac{1}{\rho_{\mathbf K}} \sum_{ \substack{0\ne I \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(I) \leq x\\ (I, P(\Omega_f))=1}} \frac{1}{{\mathfrak N}(I)} - \delta_{\mathbf K}(\Omega_f) \log x \bigg) \end{align*} $$

exists and is denoted by $\gamma _{\mathbf K}(\Omega _f)$ .

Proof. Let $\Omega _f \subset \mathcal {P}_{\mathbf K}$ and ${\mathfrak {p}} \in \mathcal {P}_{\mathbf K}$ be a prime ideal not in $\Omega _f$ . Using

$$ \begin{align*} \sum_{\substack{0\neq I\subseteq {{\mathcal{O}}}_{\mathbf K} \\ {\mathfrak N}(I)\leq x \\ (I,{\mathfrak{p}} P(\Omega_f) )=1}} \frac{1}{{\mathfrak N}(I)}= \sum_{\substack{0\neq I\subseteq {{\mathcal{O}}}_{\mathbf K} \\ {\mathfrak N}(I)\leq x \\ (I,P(\Omega_f))=1}} \frac{1}{{\mathfrak N}(I)}-\frac{1}{{\mathfrak N}({\mathfrak{p}})} \sum_{\substack{0\neq I\subseteq {{\mathcal{O}}}_{\mathbf K} \\ {\mathfrak N}(I)\leq {x}/{{\mathfrak N}({\mathfrak{p}})} \\ (I,P(\Omega_f))=1}}\frac{1}{{\mathfrak N}(I)}, \end{align*} $$

the result follows by induction on the cardinality of $\Omega _f$ .

Lemma 3.2. Let $\Omega _f$ be a finite set of nonzero prime ideals. Then,

$$ \begin{align*} \gamma_{\mathbf K}(\Omega_f) = \delta_{\mathbf K}(\Omega_f) \bigg( \gamma_{\mathbf K} + \sum_{ \mathfrak{p}\in \Omega_f} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} \bigg). \end{align*} $$

Proof. We have

$$ \begin{align*} \sum_{ \substack{0\ne I \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(I)\leq x \\ (I, P(\Omega_f)) = 1}} \frac{1}{{\mathfrak N}(I)} &= \sum_{ \substack{0\ne I \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(I) \leq x}} \frac{1}{{\mathfrak N}(I)} \sum_{J\mid(I,P(\Omega_f))} \mu(J)\\ & =\sum_{J\mid P(\Omega_f)} \frac{\mu(J)}{{\mathfrak N}(J)} \sum_{ \substack{0\ne J_0 \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(J_0) \leq {x}/{{\mathfrak N}(J)}}} \frac{1}{{\mathfrak N}(J_0)}\\ &=\sum_{J \mid P(\Omega_f)} \frac{\mu(J)}{{\mathfrak N}(J)} \left\{\rho_{\mathbf K} \log \frac{x}{{\mathfrak N}(J)} +\rho_{{\mathbf K}} \gamma_{\mathbf K} + \mathrm{o}(1)\right\}\\ &= \delta_{\mathbf K}(\Omega_f) \big( \rho_{\mathbf K} \log x + \rho_{{\mathbf K}}\gamma_{\mathbf K} + \mathrm{o}(1)\big) - \rho_{\mathbf K}\sum_{J\mid P(\Omega_f)} \frac{\mu(J)}{{\mathfrak N}(J)} \log {\mathfrak N}(J). \end{align*} $$

We now consider the last term:

$$ \begin{align*} -\sum_{J\mid P(\Omega_f)} \frac{\mu(J)}{{\mathfrak N}(J)} \log {\mathfrak N}(J) &= \sum_{J \mid P(\Omega_f)} \frac{1}{{\mathfrak N}(J)} \sum_{J_0\mid J} \Lambda(J_0) \mu(JJ_0^{-1})\\&= \sum_{J_0 \mid P(\Omega_f)} \frac{\Lambda(J_0)}{ {\mathfrak N}(J_0)} \sum_{J_1\mid P(\Omega_f)J_0^{-1}} \frac{\mu(J_1)}{{\mathfrak N}(J_1)}\\& = \sum_{\mathfrak{p'} \in \Omega_f} \frac{\Lambda(\mathfrak{p'})}{ {\mathfrak N}(\mathfrak{p'})} \sum_{J_1\mid P(\Omega_f)\mathfrak{p'}^{-1}} \frac{\mu(J_1)}{{\mathfrak N}(J_1)}\\&= \sum_{\mathfrak{p'} \in \Omega_f} \frac{\log{\mathfrak N}(\mathfrak{p'})}{ {\mathfrak N}(\mathfrak{p'})}\bigg( \frac{\delta_{\mathbf K}(\Omega_f)}{1- {1}/{{\mathfrak N}(\mathfrak{p'})}}\bigg)\\& = \delta_{\mathbf K}(\Omega_f) \sum_{\mathfrak{p} \in \Omega_f} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1}. \end{align*} $$

Thus,

$$ \begin{align*} \lim_{x\rightarrow \infty} \bigg(\frac{1}{\rho_{\mathbf K}} \sum_{ \substack{0\ne I \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(I)\leq x \\ (I, P(\Omega_f)) = 1}} \frac{1}{{\mathfrak N}(I)} - \delta_{\mathbf K}(\Omega_f) \log x \bigg) &= \delta_{\mathbf K}(\Omega_f) \bigg( \gamma_{\mathbf K} + \sum_{ \mathfrak{p}\in \Omega_f} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} \bigg).\\[-60.5pt] \end{align*} $$

Corollary 3.3. For a number field ${\mathbf K}$ and any set $\Omega \subset \mathcal {P}_{\mathbf K}$ satisfying (1.2), the limit

$$ \begin{align*} \lim_{ x \to \infty} \bigg( \frac{1}{\rho_{\mathbf K}} \sum_{ \substack{0\ne I \subset {{\mathcal{O}}}_{{\mathbf K}} \\ {\mathfrak N}(I) \leq x \\ (I, P(\Omega(x)))=1}} \frac{1}{{\mathfrak N}(I)} - \delta_{\mathbf K}(\Omega(x)) \log x \bigg) \end{align*} $$

exists and equals

$$ \begin{align*} \delta_{\mathbf K}(\Omega) \bigg( \gamma_{\mathbf K} + \sum_{ \mathfrak{p}\in \Omega} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} \bigg). \end{align*} $$

We denote this limit by $\gamma _{\mathbf K}(\Omega )$ .

Proof. Follows from Lemma 3.2 since $\Omega (x)$ is a finite set.

4 Proof of Theorem 1.2

Suppose there exist $i,j \in I$ such that

$$ \begin{align*} \frac{\gamma_{\mathbf K}(\Omega_i)}{\delta_{\mathbf K}(\Omega_i)} \quad\text{and} \quad \frac{\gamma_{\mathbf K}(\Omega_j)}{ \delta_{\mathbf K}(\Omega_j)} \end{align*} $$

are algebraic. Using Corollary 3.3,

(4.1) $$ \begin{align} \frac{\gamma_{\mathbf K}(\Omega_i)}{\delta_{\mathbf K}(\Omega_i)} - \frac{\gamma_{\mathbf K}(\Omega_j)} {\delta_{\mathbf K}(\Omega_j)} &= \sum_{ \mathfrak{p}\in \Omega_i} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} -\sum_{ \mathfrak{p}\in \Omega_j} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1}, \end{align} $$

which is also an algebraic number. Since the sets $N_{\Omega _i} \backslash N_{\Omega _j}$ and $N_{\Omega _i} \backslash N_{\Omega _j}$ are nonempty and finite, the sets $\Omega _i \backslash \Omega _j$ and $\Omega _j \backslash \Omega _i$ are also finite. Let

$$ \begin{align*} \Omega_i \backslash \Omega_j= \{\mathfrak{p}_1, \mathfrak{p}_2 , \ldots, \mathfrak{p}_n\}, \quad \Omega_j \backslash \Omega_i= \{\mathfrak{q}_1, \mathfrak{q}_2 , \ldots, \mathfrak{q}_m\}. \end{align*} $$

Then (4.1) implies

(4.2) $$ \begin{align} \sum_{ \mathfrak{p}\in \Omega_i} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} - \sum_{ \mathfrak{p}\in \Omega_j} \frac{\log {\mathfrak N}(\mathfrak{p})}{{\mathfrak N}(\mathfrak{p}) - 1} = \sum_{s = 1}^{n} \frac {\log p^{f_s}_s}{ p^{f_s}_s - 1} - \sum_{t=1}^{m} \frac {\log q^{g_t}_t}{ q^{g_t}_t - 1} = \log \Bigg( \frac{\prod_{s=1}^{n} p^{({f_s}/{p^{f_s}_s -1})}_s} {\prod_{t=1}^{m} q^{({g_t}/{q^{g_t}_t -1})}_t}\Bigg), \end{align} $$

where ${\mathfrak N}(\mathfrak {p}_s)=p_s^{f_s}$ and ${\mathfrak N}(\mathfrak {q}_t)=q_t^{g_t}$ . Using Lemma 2.3 and unique prime factorisation of natural numbers, the expression in (4.2) becomes a transcendental number, which gives a contradiction.