1 Introduction and preliminaries
In 1740, Euler [Reference Euler2] introduced the Euler–Mascheroni constant, which is defined as
$$ \begin{align} \gamma = \lim_{x \to \infty} \bigg( \sum_{n \le x } \frac{1}{n} - \log x \bigg). \end{align} $$
This constant has been extensively studied (see [Reference Lagarias4]), but many questions about its behaviour are unanswered. For example, it is not known if
$\gamma $
is rational or irrational. Diamond and Ford [Reference Diamond and Ford1] 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, [Reference Murty and Zaytseva8]).
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 [Reference Ihara3] 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
$$ \begin{align} \sum_{\mathfrak{p}\in \Omega} \frac {\log {\mathfrak N}(\mathfrak{p})}{{\mathfrak N}(\mathfrak{p})- 1} < \infty. \end{align} $$
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 [Reference Rosen9] 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 [Reference Murty and Esmonde7, 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 [Reference Murty6, 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, [Reference Lindemann5]).
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,
$$ \begin{align} \frac{\gamma_{\mathbf K}(\Omega_i)}{\delta_{\mathbf K}(\Omega_i)} - \frac{\gamma_{\mathbf K}(\Omega_j)} {\delta_{\mathbf K}(\Omega_j)} &= \sum_{ \mathfrak{p}\in \Omega_i} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} -\sum_{ \mathfrak{p}\in \Omega_j} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1}, \end{align} $$
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
$$ \begin{align} \sum_{ \mathfrak{p}\in \Omega_i} \frac {\log {\mathfrak N}(\mathfrak{p})}{ {\mathfrak N}(\mathfrak{p}) - 1} - \sum_{ \mathfrak{p}\in \Omega_j} \frac{\log {\mathfrak N}(\mathfrak{p})}{{\mathfrak N}(\mathfrak{p}) - 1} = \sum_{s = 1}^{n} \frac {\log p^{f_s}_s}{ p^{f_s}_s - 1} - \sum_{t=1}^{m} \frac {\log q^{g_t}_t}{ q^{g_t}_t - 1} = \log \Bigg( \frac{\prod_{s=1}^{n} p^{({f_s}/{p^{f_s}_s -1})}_s} {\prod_{t=1}^{m} q^{({g_t}/{q^{g_t}_t -1})}_t}\Bigg), \end{align} $$
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.
Acknowledgements
The authors would like to thank Prof. S. Gun for suggesting the problem and IMSc for providing academic facilities. They would also like to thank Prof. P. Moree and J. Sivaraman for comments on an earlier version of the manuscript, which improved the exposition. The authors would like to thank the anonymous referee for careful reading of the paper and helpful comments. The first author would like to thank the Max-Planck-Institut für Mathematik for providing a friendly atmosphere.
