The Euler–Mascheroni constant γ≈0.57722\gamma\approx 0.57722 and the Euler–Gompertz constant δ≈0.59635\delta\approx 0.59635 are not individually known to be irrational, yet Hardy's identity γ+δ/e=Ein(1)\gamma+\delta/e=\mathrm{Ein}(1) exhibits one transcendental point on the real line of linear combinations γ+αδ\gamma+\alpha\delta, since Ein\mathrm{Ein} is an E-function covered by the Siegel–Shidlovsky theorem as made explicit by Rivoal (2012). Powers (2025, arXiv:2510.00315) proves that α=1/e\alpha=1/e is the unique coefficient for which canonical Borel-summable divergent series for γ\gamma and δ\delta combine to an ordinarily convergent series, and states that the arithmetic nature of γ+αδ\gamma+\alpha\delta remains unknown for every real α≠1/e\alpha\neq 1/e. This record poses that question: for every real α≠1/e\alpha\neq 1/e — most notably α=0\alpha=0, where γ+αδ\gamma+\alpha\delta is Euler's constant itself — determine whether γ+αδ\gamma+\alpha\delta is rational, algebraic irrational, or transcendental. The question subsumes the irrationality of Euler's constant, one of the oldest open problems in number theory. No published work establishes the arithmetic nature of γ+αδ\gamma+\alpha\delta for any explicit α≠1/e\alpha\neq 1/e, so the problem is open.

Background

Two classical real constants sit at the boundary of what transcendence theory can currently reach. The Euler–Mascheroni constant is γ=lim⁡n→∞(1+12+⋯+1n−ln⁡n)≈0.57722\gamma=\lim_{n\to\infty}(1+\tfrac12+\cdots+\tfrac1n-\ln n)\approx 0.57722, the constant term in the asymptotics of the harmonic numbers and equivalently γ=−Γ′(1)\gamma=-\Gamma'(1), minus the logarithmic derivative of the Gamma function at 11. The Euler–Gompertz constant is δ=∫0∞e−t/(1+t) dt≈0.59635\delta=\int_0^\infty e^{-t}/(1+t)\,dt\approx 0.59635, which equals −e Ei(−1)-e\,\mathrm{Ei}(-1) for the exponential integral Ei\mathrm{Ei}; it appears in Ramanujan's first letter to Hardy and in the theory of summation of divergent series (Hardy, Divergent Series, 1949). Neither number is known to be irrational; for γ\gamma this has been open since Euler computed the constant in the 1730s (Lagarias, 2013). A well-known equation recorded by Hardy links the two constants through Ein(z)=∑k≥1(−1)k+1zk/(k k!)=∫0z(1−e−t)/t dt\mathrm{Ein}(z)=\sum_{k\ge1}(-1)^{k+1}z^k/(k\,k!)=\int_0^z(1-e^{-t})/t\,dt, an entire-function analog of the exponential integral, namely γ+δ/e=Ein(1)\gamma+\delta/e=\mathrm{Ein}(1). The function Ein\mathrm{Ein} is an E-function in Siegel's sense: an entire function ∑anzn/n!\sum a_n z^n/n! with algebraic coefficients of arithmetically controlled size satisfying a linear differential equation with rational-function coefficients (here z Ein′(z)=1−e−zz\,\mathrm{Ein}'(z)=1-e^{-z}). The Siegel–Shidlovsky theorem states that algebraic-independence relations over Q(z)\mathbb{Q}(z) among such functions persist at nonzero algebraic arguments, and Beukers' refined version removes the homogeneity restriction. Applied to the pair (Ein,e−z)(\mathrm{Ein},e^{-z}), this proves Ein(1)=γ+δ/e\mathrm{Ein}(1)=\gamma+\delta/e transcendental (as made explicit by Rivoal), so one point of the real line {γ+αδ:α∈R}\{\gamma+\alpha\delta:\alpha\in\mathbb{R}\} is pinned down. Rivoal (2012) obtained the strongest joint information on the pair: with G0(z)=∫0∞e−t/(t+z) dtG_0(z)=\int_0^\infty e^{-t}/(t+z)\,dt, his Theorem 1(ii) gives the quantitative simultaneous-approximation bound ∣γ+log⁡z−p/q∣+∣G0(z)−r/q∣≥d H−3−ε|\gamma+\log z-p/q|+|G_0(z)-r/q|\ge d\,H^{-3-\varepsilon} for rational z>0z>0 (so at least one of γ,δ\gamma,\delta is irrational, a disjunctive statement first noted by Aptekarev), and his Theorem 2(ii) shows the field generated by eze^z, γ+log⁡z\gamma+\log z and G0(z)G_0(z) has transcendence degree at least 22 over Q\mathbb{Q} (contains at least two algebraically independent numbers), so at least one of γ\gamma and δ\delta is transcendental. Powers (2025) approaches the line γ+αδ\gamma+\alpha\delta analytically. Writing both constants as Laplace integrals and expanding the kernels term by term yields canonical Borel-summable divergent series; Borel summation assigns to a suitable divergent series ∑ak\sum a_k the value ∫0∞B(u)e−udu\int_0^\infty\mathcal{B}(u)e^{-u}du, where B(u)=∑akuk/k!\mathcal{B}(u)=\sum a_k u^k/k! is its Borel transform. For γ\gamma this gives Sγ=∑k≥1(−1)k(!k)/kS_\gamma=\sum_{k\ge1}(-1)^k(!k)/k, with !k!k the kk-th derangement number !k=k!∑ℓ=0k(−1)ℓ/ℓ!!k=k!\sum_{\ell=0}^k(-1)^\ell/\ell!, and for δ\delta the classical Euler–Ramanujan series Sδ=∑k≥1(−1)k−1(k−1)!S_\delta=\sum_{k\ge1}(-1)^{k-1}(k-1)!. Powers' Proposition 1 shows that the combined series Sγ+αSδS_\gamma+\alpha S_\delta converges in the ordinary sense if and only if α=1/e\alpha=1/e, and the same uniqueness persists for the generalizations γ(n)+αδ(n)\gamma^{(n)}+\alpha\delta^{(n)} built from moments of the Gumbel(0,1)(0,1) distribution, the probability law with density e−x−e−xe^{-x-e^{-x}} whose mean is exactly γ\gamma; a companion preprint (Powers, 2025) claims the values γ(n)+δ(n)/e\gamma^{(n)}+\delta^{(n)}/e are transcendental for all n≥0n\ge 0 on this α=1/e\alpha=1/e ray. In parallel, the rational-approximation program on γ\gamma and δ\delta separately — Hermite–Padé-type simultaneous rational approximants, constructed from multiple orthogonal polynomials — polynomials satisfying orthogonality relations against several weight functions simultaneously rather than a single one — for the function system (1,exp⁡)(1,\exp) (Aptekarev-type constructions, Rivoal, and most recently Wolfs–Van Assche 2025), and continued-fraction expansions (Hessami Pilehrood and Hessami Pilehrood, 2013) — has steadily improved approximation quality but still falls short of proving either constant irrational. What no current method reaches is the arithmetic nature of γ+αδ\gamma+\alpha\delta at any coefficient other than the single transcendental point α=1/e\alpha=1/e.

Problem Statement

For every real α≠1/e\alpha\neq 1/e, determine the arithmetic nature of γ+αδ\gamma+\alpha\delta — rational, algebraic irrational, or transcendental — where γ=lim⁡n→∞(1+12+⋯+1n−ln⁡n)≈0.57722\gamma=\lim_{n\to\infty}(1+\tfrac12+\cdots+\tfrac1n-\ln n)\approx 0.57722 is the Euler–Mascheroni constant and δ=∫0∞e−t/(1+t) dt≈0.59635\delta=\int_0^\infty e^{-t}/(1+t)\,dt\approx 0.59635 is the Euler–Gompertz constant. The one settled point is α=1/e\alpha=1/e: γ+δ/e=Ein(1)\gamma+\delta/e=\mathrm{Ein}(1) is transcendental. A complete solution determines the trichotomy for every real α≠1/e\alpha\neq 1/e. Two nonconstructive facts delimit but do not answer the question: trivially, γ+αδ\gamma+\alpha\delta is transcendental for all but countably many real α\alpha (without identifying a single one), and Rivoal's theorem that at least one of γ\gamma and δ\delta is transcendental guarantees that some explicit α≠1/e\alpha\neq 1/e works without indicating which. No explicitly given real α≠1/e\alpha\neq 1/e is known for which γ+αδ\gamma+\alpha\delta is provably irrational.

References

  1. Powers, Michael R., 2025. On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants. arXiv:2510.00315, DOI 10.48550/arXiv.2510.00315. https://arxiv.org/abs/2510.00315
  2. Powers, Michael R., 2025. On the Algebraic Independence of a Set of Generalized Constants. arXiv:2508.12123, DOI 10.48550/arXiv.2508.12123. https://arxiv.org/abs/2508.12123
  3. Rivoal, Tanguy, 2012. On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant. Michigan Mathematical Journal 61(2), 239–254. DOI 10.1307/mmj/1339011525. https://doi.org/10.1307/mmj/1339011525
  4. Beukers, F., 2006. A refined version of the Siegel–Shidlovskii theorem. Annals of Mathematics 163(1), 369–379. DOI 10.4007/annals.2006.163.369. https://doi.org/10.4007/annals.2006.163.369
  5. Lagarias, J. C., 2013. Euler's constant: Euler's work and modern developments. Bulletin of the American Mathematical Society 50(4), 527–628. DOI 10.1090/s0273-0979-2013-01423-x. https://doi.org/10.1090/s0273-0979-2013-01423-x
  6. Hardy, G. H., 1949. Divergent Series. Clarendon Press, Oxford. zbMATH Zbl 0032.05801. https://zbmath.org/?q=an%3A0032.05801
  7. Van Assche, Walter; Wolfs, Thomas, 2025. Rational approximation of Euler's constant using multiple orthogonal polynomials. Combinatorics and Number Theory 14(2), 141–162. DOI 10.2140/cnt.2025.14.141; also arXiv:2404.09799. https://doi.org/10.2140/cnt.2025.14.141
  8. Hessami Pilehrood, Kh.; Hessami Pilehrood, T., 2013. On a continued fraction expansion for Euler's constant. Journal of Number Theory 133(2), 769–786. DOI 10.1016/j.jnt.2012.08.016. https://doi.org/10.1016/j.jnt.2012.08.016
  9. Bolbachan, Vasily, 2011. Rational approximants for the Euler-Gompertz constant. arXiv:1104.4721, DOI 10.48550/arXiv.1104.4721. https://arxiv.org/abs/1104.4721