For a number field K\mathbf K, the Euler–Kronecker constant γK\gamma_{\mathbf K} is the ratio of the constant term to the residue of the Dedekind zeta function ζK(s)\zeta_{\mathbf K}(s) at s=1s=1, generalizing Euler's constant γ=γQ\gamma=\gamma_{\mathbb Q}. Kandhil and Lunia introduced generalized constants γK(Ω)\gamma_{\mathbf K}(\Omega), defined by sums over integral ideals coprime to a set Ω\Omega of prime ideals, and proved that in any admissible family {Ωi}\{\Omega_i\} at most one normalized constant γK(Ωi)/δK(Ωi)\gamma_{\mathbf K}(\Omega_i)/\delta_{\mathbf K}(\Omega_i) is algebraic. Their identity γK(Ω)/δK(Ω)=γK+∑p∈Ωlog⁡N(p)/(N(p)−1)\gamma_{\mathbf K}(\Omega)/\delta_{\mathbf K}(\Omega)=\gamma_{\mathbf K}+\sum_{\mathfrak p\in\Omega}\log\mathfrak N(\mathfrak p)/(\mathfrak N(\mathfrak p)-1) and Lindemann's theorem classify only differences between constants, never a single one. The open problem is to determine, for a specific number field K\mathbf K and admissible Ω\Omega, whether γK(Ω)/δK(Ω)\gamma_{\mathbf K}(\Omega)/\delta_{\mathbf K}(\Omega) is algebraic or transcendental - beginning with Ω=∅\Omega=\emptyset, where for K=Q\mathbf K=\mathbb Q this is the classical, still-open question of the irrationality of Euler's constant.

Background

Euler's constant γ=lim⁡x→∞(∑n≤x1/n−log⁡x)\gamma=\lim_{x\to\infty}(\sum_{n\le x}1/n-\log x), introduced by Euler in 1740, is the constant term in the Laurent expansion (the power-series expansion around a pole) of the Riemann zeta function ζ(s)\zeta(s) at s=1s=1. Its basic arithmetic nature is famously unresolved: it is not known whether γ\gamma is rational or irrational (Lagarias, 2013).

A number field K\mathbf K is a finite-degree extension of Q\mathbb Q; its ring of integers OK\mathcal O_{\mathbf K} consists of the elements of K\mathbf K satisfying a monic polynomial with integer coefficients. Nonzero prime ideals p\mathfrak p of OK\mathcal O_{\mathbf K} are the multiplicative building blocks of its ideals, and the norm N(a)=[OK:a]\mathfrak N(\mathfrak a)=[\mathcal O_{\mathbf K}:\mathfrak a] of a nonzero integral ideal a\mathfrak a is a positive integer. The Dedekind zeta function ζK(s)=∑aN(a)−s\zeta_{\mathbf K}(s)=\sum_{\mathfrak a}\mathfrak N(\mathfrak a)^{-s}, summed over nonzero integral ideals, has a simple pole at s=1s=1 with residue ρK\rho_{\mathbf K}. Ihara (2009) defined the Euler–Kronecker constant of K\mathbf K as γK:=cK/ρK\gamma_{\mathbf K}:=c_{\mathbf K}/\rho_{\mathbf K}, where cKc_{\mathbf K} is the constant term of the Laurent expansion ζK(s)=ρK/(s−1)+cK+O(s−1)\zeta_{\mathbf K}(s)=\rho_{\mathbf K}/(s-1)+c_{\mathbf K}+O(s-1); equivalently γK\gamma_{\mathbf K} is the constant term of ζK′/ζK\zeta'_{\mathbf K}/\zeta_{\mathbf K} near s=1s=1 (that is, ζK′/ζK=−1/(s−1)+γK+O(s−1)\zeta'_{\mathbf K}/\zeta_{\mathbf K}=-1/(s-1)+\gamma_{\mathbf K}+O(s-1)). This invariant governs Mertens-type theorems for prime ideals (the prime-ideal analogue of the classical asymptotic ∑p≤xlog⁡p∼x\sum_{p\le x}\log p\sim x for rational primes) and Brauer–Siegel-type behavior (the classical asymptotic relationship linking the class number, regulator (a volume measuring the size of the units of OK\mathcal O_{\mathbf K}), and discriminant of K\mathbf K), and γQ=γ\gamma_{\mathbb Q}=\gamma. For abelian K\mathbf K, ζK\zeta_{\mathbf K} factors into Dirichlet LL-functions (series attached to multiplicative characters), and γK=γ+∑χ≠χ0L′(1,χ)/L(1,χ)\gamma_{\mathbf K}=\gamma+\sum_{\chi\ne\chi_0}L'(1,\chi)/L(1,\chi), summed over the nontrivial characters χ\chi of Gal(K/Q)\mathrm{Gal}(\mathbf K/\mathbb Q), so these constants couple γ\gamma with logarithmic derivatives of LL-functions, a class of values for which no general transcendence theory exists.

Diamond and Ford (2008) introduced, for a finite set Ω\Omega of rational primes, generalized Euler constants γ(Ω)\gamma(\Omega) (the K=Q\mathbf K=\mathbb Q case of the construction below), and Murty and Zaytseva (2013) proved that at most one number in the infinite list {γ(Ω)}\{\gamma(\Omega)\} is algebraic, without identifying any member. Kandhil and Lunia (2024) extended the construction to number fields: for a (possibly infinite) set Ω\Omega of nonzero prime ideals of OK\mathcal O_{\mathbf K} satisfying the convergence condition ∑p∈Ωlog⁡N(p)/(N(p)−1)<∞\sum_{\mathfrak p\in\Omega}\log\mathfrak N(\mathfrak p)/(\mathfrak N(\mathfrak p)-1)<\infty, let δK(Ω)=∏p∈Ω(1−1/N(p))\delta_{\mathbf K}(\Omega)=\prod_{\mathfrak p\in\Omega}(1-1/\mathfrak N(\mathfrak p)) and define

γK(Ω)=lim⁡x→∞(1ρK∑0≠a⊂OKN(a)≤x(a,P(Ω(x)))=11N(a)−δK(Ω(x))log⁡x),\gamma_{\mathbf K}(\Omega)=\lim_{x\to\infty}\Big(\frac{1}{\rho_{\mathbf K}}\sum_{\substack{0\ne\mathfrak a\subset\mathcal O_{\mathbf K}\\ \mathfrak N(\mathfrak a)\le x\\ (\mathfrak a,P(\Omega(x)))=1}}\frac{1}{\mathfrak N(\mathfrak a)}-\delta_{\mathbf K}(\Omega(x))\log x\Big),

where P(Ω(x))P(\Omega(x)) is the product of the prime ideals in Ω\Omega of norm at most xx, the condition (a,P(Ω(x)))=1(\mathfrak a,P(\Omega(x)))=1 means a\mathfrak a is coprime to that product, and Ω(x)\Omega(x) denotes the prime ideals in Ω\Omega of norm at most xx. The limit exists, and γK(∅)=γK\gamma_{\mathbf K}(\emptyset)=\gamma_{\mathbf K}. Their key identity (Corollary 3.3) is

γK(Ω)δK(Ω)=γK+∑p∈Ωlog⁡N(p)N(p)−1.\frac{\gamma_{\mathbf K}(\Omega)}{\delta_{\mathbf K}(\Omega)}=\gamma_{\mathbf K}+\sum_{\mathfrak p\in\Omega}\frac{\log\mathfrak N(\mathfrak p)}{\mathfrak N(\mathfrak p)-1}.

Their Theorem 1.2 states: if {Ωi}i∈I\{\Omega_i\}_{i\in I} is a family of such sets for which NΩi∖NΩjN_{\Omega_i}\setminus N_{\Omega_j} is nonempty and finite for all i≠ji\ne j (where NΩN_{\Omega} is the set of rational primes lying below the prime ideals of Ω\Omega), then at most one of the normalized constants γK(Ωi)/δK(Ωi)\gamma_{\mathbf K}(\Omega_i)/\delta_{\mathbf K}(\Omega_i) is algebraic. The proof rests on Lindemann's 1882 theorem that log⁡α\log\alpha is transcendental for every algebraic α≠0,1\alpha\ne 0,1: if two normalized constants in the family were algebraic, their difference would be a nonzero rational linear combination of logarithms of prime powers, hence transcendental - a contradiction.

The method therefore classifies only differences between members of a family; by construction it cannot decide the status of any single constant. For a finite nonempty Ω\Omega, the sum term, after grouping its terms by the underlying rational prime pp (using log⁡N(p)=fplog⁡p\log\mathfrak N(\mathfrak p)=f_{\mathfrak p}\log p, where fpf_{\mathfrak p} is the residue degree of p\mathfrak p), is itself a nonzero rational linear combination of logarithms of distinct rational primes, hence transcendental, so the algebraicity question for every member is inseparably tied to the unknown arithmetic nature of γK\gamma_{\mathbf K} itself. For K=Q\mathbf K=\mathbb Q and Ω=∅\Omega=\emptyset, that is exactly the centuries-old rationality question for Euler's constant.

Problem Statement

Let K\mathbf K be a number field, that is, a finite-degree extension of Q\mathbb Q, with ring of integers OK\mathcal O_{\mathbf K} (the elements of K\mathbf K satisfying a monic polynomial with integer coefficients); for a nonzero integral ideal a⊆OK\mathfrak a\subseteq\mathcal O_{\mathbf K} write N(a)=[OK:a]\mathfrak N(\mathfrak a)=[\mathcal O_{\mathbf K}:\mathfrak a] for its norm, a positive integer. Let ζK(s)=∑aN(a)−s\zeta_{\mathbf K}(s)=\sum_{\mathfrak a}\mathfrak N(\mathfrak a)^{-s}, summed over the nonzero integral ideals of OK\mathcal O_{\mathbf K}, be the Dedekind zeta function of K\mathbf K, with Laurent expansion ζK(s)=ρK/(s−1)+cK+O(s−1)\zeta_{\mathbf K}(s)=\rho_{\mathbf K}/(s-1)+c_{\mathbf K}+O(s-1) at s=1s=1, and let γK=cK/ρK\gamma_{\mathbf K}=c_{\mathbf K}/\rho_{\mathbf K} be the Euler–Kronecker constant of K\mathbf K; for K=Q\mathbf K=\mathbb Q this is Euler's constant γ\gamma. Let Ω\Omega be a set of nonzero prime ideals of OK\mathcal O_{\mathbf K} satisfying the convergence condition ∑p∈Ωlog⁡N(p)/(N(p)−1)<∞\sum_{\mathfrak p\in\Omega}\log\mathfrak N(\mathfrak p)/(\mathfrak N(\mathfrak p)-1)<\infty, and let δK(Ω)=∏p∈Ω(1−1/N(p))\delta_{\mathbf K}(\Omega)=\prod_{\mathfrak p\in\Omega}(1-1/\mathfrak N(\mathfrak p)). The generalized Euler–Kronecker constant γK(Ω)\gamma_{\mathbf K}(\Omega) attached to the pair (K,Ω)(\mathbf K,\Omega) (defined by the limiting procedure recalled in the Background) satisfies the identity

γK(Ω)δK(Ω)=γK+∑p∈Ωlog⁡N(p)N(p)−1\frac{\gamma_{\mathbf K}(\Omega)}{\delta_{\mathbf K}(\Omega)}=\gamma_{\mathbf K}+\sum_{\mathfrak p\in\Omega}\frac{\log\mathfrak N(\mathfrak p)}{\mathfrak N(\mathfrak p)-1}

(Corollary 3.3 of Kandhil and Lunia). Determine the arithmetic nature of this quantity for specific instances: is it algebraic, or is it transcendental?

In particular: (1) for K=Q\mathbf K=\mathbb Q and Ω=∅\Omega=\emptyset, is Euler's constant γ\gamma rational, algebraic irrational, or transcendental - it is not even known whether γ\gamma is irrational, and an irrationality proof alone would leave the algebraic-versus-transcendental alternative open; (2) for a single fixed number field K≠Q\mathbf K\ne\mathbb Q (for example K=Q(i)\mathbf K=\mathbb Q(i)), is γK\gamma_{\mathbf K} transcendental? and (3) for a specified member Ωj\Omega_j of an admissible family {Ωi}\{\Omega_i\} as in Theorem 1.2 of Kandhil and Lunia, is γK(Ωj)/δK(Ωj)\gamma_{\mathbf K}(\Omega_j)/\delta_{\mathbf K}(\Omega_j) algebraic or transcendental (the instance-level form of the family question of which member, if any, is the unique possible algebraic one)? Each of these instances is open: Kandhil and Lunia's theorem establishes only that at most one member of an admissible family is algebraic, without identifying which one.

References

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