The Euler–Gompertz constant δ=∫0∞e−x/(1+x) dx=eE1(1)≈0.5963473623…\delta=\int_0^\infty e^{-x}/(1+x)\,dx=eE_1(1)\approx 0.5963473623\ldots is a named classical constant of analysis whose individual arithmetic nature is unknown: it is not known whether δ\delta is rational, algebraic irrational, or transcendental. What is known is only disjunctive: by work of Mahler, reproved by Rivoal, at least one of Euler's constant γ\gamma and δ\delta is transcendental, and the pair (γ,δ)(\gamma,\delta) admits a simultaneous rational-approximation measure; generalized disjunctive results and a conditional irrationality criterion for δ\delta have appeared up to 2025. The problem is to determine the arithmetic nature of δ\delta itself — at minimum, to prove that δ\delta is irrational, or to prove that it is rational. The literature through 2025–2026 uniformly treats this as open; the shared Padé/continued-fraction machinery that produces linear forms in 11 and δ\delta converges only sub-exponentially and provably falls short of the classical irrationality criterion.

Background

Two constants of classical analysis have famously unresolved arithmetic status. Euler's constant is γ=lim⁡n→∞(1+1/2+⋯+1/n−log⁡n)≈0.5772156649…\gamma=\lim_{n\to\infty}(1+1/2+\cdots+1/n-\log n)\approx 0.5772156649\ldots, and the Euler–Gompertz constant is δ=∫0∞e−x1+x dx=∫0∞log⁡(1+x) e−x dx≈0.5963473623….\delta=\int_0^\infty\frac{e^{-x}}{1+x}\,dx=\int_0^\infty\log(1+x)\,e^{-x}\,dx\approx 0.5963473623\ldots. The two are tied by Hardy's formula δ=−e(γ+∑n≥1(−1)nn⋅n!)\delta=-e\bigl(\gamma+\sum_{n\ge 1}\frac{(-1)^n}{n\cdot n!}\bigr), so that δ=eE1(1)\delta=eE_1(1), where E1(z)=∫z∞e−t dt/tE_1(z)=\int_z^\infty e^{-t}\,dt/t is the exponential integral — a function with a logarithmic singularity at the origin. The constant δ\delta arose in Gompertz's work on mortality models and in Stieltjes's 1895 theory of continued fractions, where the expansion δ=12+Km=1∞(−m22(m+1))=12−124−226−328−⋯\delta=\tfrac12+\mathbf K_{m=1}^\infty\bigl(-\tfrac{m^2}{2(m+1)}\bigr)=\cfrac{1}{2-\cfrac{1^2}{4-\cfrac{2^2}{6-\cfrac{3^2}{8-\cdots}}}} appears (here K\mathbf K denotes a generalized continued fraction in Wall's classical notation, in which the leading 12\tfrac12 stands for 1/(2+⋯ )1/(2+\cdots), so that the whole expression equals the nested fraction displayed; reproduced by Pilehrood and Pilehrood through a second-order recurrence with polynomial coefficients). The irrationality of neither γ\gamma nor δ\delta is known; both problems sit just beyond the reach of the standard irrationality engine that succeeded for ζ(3)=∑n≥11/n3\zeta(3)=\sum_{n\ge 1}1/n^3 in 1978, namely explicit sequences of linear forms unα−vnu_n\alpha-v_n built from Padé approximants (rational functions matching a function's series expansion to maximal order) or hypergeometric integrals, combined with an integrality-versus-decay argument. For δ\delta this machinery is unusually close to sufficient: the convergents sn/qns_n/q_n of Stieltjes's continued fraction satisfy δ−sn/qn=e−4n (2πe+O(n−1/2))\delta-s_n/q_n=e^{-4\sqrt n}\,(2\pi e+O(n^{-1/2})), and the same recurrence governs recent rational approximations to γ\gamma, yet the sub-exponential decay is too slow relative to the growth of the denominators qnq_n, exactly as for γ\gamma. On the transcendence side, Siegel's EE-functions (power series ∑anzn/n!\sum a_n z^n/n! with algebraic coefficients and controlled growth, e.g. eze^z) obey the Siegel–Shidlovsky theorem, giving algebraic-independence statements for their values at algebraic points; Mahler used it to prove that ee and ∑k≥1(−1)kk k!\sum_{k\ge 1}\frac{(-1)^k}{k\,k!} are algebraically independent, which through Hardy's formula yields only the disjunctive conclusion that at least one of γ,δ\gamma,\delta is transcendental. Because exE1(x)e^xE_1(x) is not an EE-function (the logarithmic singularity blocks this), the individual value δ\delta escapes the theory: the obstacle is genuinely arithmetic, not analytic. Rivoal strengthened the picture with a new proof of the disjunctive result and a measure showing that the pair (γ,δ)(\gamma,\delta) cannot be very well approximated simultaneously by rationals with a common denominator; Murabayashi and Yoshida later built a convergent continued fraction for exE1(x)e^xE_1(x) on x>0x>0 together with rationals Ai/BiA_i/B_i with δBi−Ai→0\delta B_i-A_i\to 0 and an explicit sufficient condition whose verification would prove δ\delta irrational; and Powers proved algebraic independence of the whole family γ(n)+δ(n)/e\gamma^{(n)}+\delta^{(n)}/e, where γ(n)\gamma^{(n)} and δ(n)\delta^{(n)} are generalized Euler constants built from the moments of the standard Gumbel (extreme-value) distribution, always leaving the individual status of δ\delta untouched. The comprehensive surveys of Lagarias record the same state of knowledge. This is the background against which the question below is asked.

Problem Statement

Determine the arithmetic nature of the Euler–Gompertz constant δ=∫0∞e−x1+x dx=eE1(1)≈0.5963473623…\delta=\int_0^\infty\frac{e^{-x}}{1+x}\,dx=eE_1(1)\approx 0.5963473623\ldots: prove that δ\delta is irrational (and, if possible, decide whether it is transcendental, as predicted by the disjunctive theorem that at least one of γ\gamma and δ\delta is transcendental), or prove that δ\delta is rational. Here δ\delta is to be understood exactly as defined, without restrictions or renormalizations.

References

  1. Kh. Hessami Pilehrood and T. Hessami Pilehrood, "On a continued fraction expansion for Euler's constant", arXiv:1010.1420 (2010), https://arxiv.org/abs/1010.1420; Journal of Number Theory 133 (2013), 769–786, DOI 10.1016/j.jnt.2012.08.016, https://doi.org/10.1016/j.jnt.2012.08.016
  2. Anne-Maria Ernvall-Hytönen, Tapani Matala-aho and Louna Seppälä, "Euler's factorial series, Hardy integral, and continued fractions", arXiv:2111.13649, https://arxiv.org/abs/2111.13649; Journal of Number Theory 244 (2023), 224–250, DOI 10.1016/j.jnt.2022.09.007, https://doi.org/10.1016/j.jnt.2022.09.007
  3. T. Rivoal, "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant", Michigan Mathematical Journal 61 (2012), no. 2, 239–254, DOI 10.1307/mmj/1339011525, https://doi.org/10.1307/mmj/1339011525
  4. K. Mahler, "Applications of a theorem by A. B. Shidlovski", Proceedings of the Royal Society of London. Series A 305 (1968), 149–173, DOI 10.1098/rspa.1968.0111, https://doi.org/10.1098/rspa.1968.0111
  5. Naoki Murabayashi and Hayato Yoshida, "Rational function approximations of the special function exE1(x)e^xE_1(x) and applications to irrationality of Euler–Gompertz constant δ\delta", arXiv:2210.06768 (2022), https://arxiv.org/abs/2210.06768 (published in extended form as "On a continued fraction expansion of the special function exE1(x)e^xE_1(x) and an explicit expression of the continued fraction convergents", The Ramanujan Journal 66 (2025), no. 2, article 41, DOI 10.1007/s11139-024-00989-1, https://doi.org/10.1007/s11139-024-00989-1)
  6. Michael R. Powers, "On the Algebraic Independence of a Set of Generalized Constants", arXiv:2508.12123 (2025), https://arxiv.org/abs/2508.12123
  7. Michael R. Powers, "On the Uniqueness of Ein(1) among Linear Combinations of the Euler–Mascheroni and Euler–Gompertz Constants", arXiv:2510.00315 (2025), https://arxiv.org/abs/2510.00315
  8. Anne-Maria Ernvall-Hytönen and Tapani Matala-aho, "Explicit estimates for the sum ∑k=0nk!(nk)2(−1)k\sum_{k=0}^{n} k!{n\choose k}^2(-1)^k", arXiv:2310.11468 (2023), https://arxiv.org/abs/2310.11468; Proceedings of the American Mathematical Society 153 (2024), no. 2, 679–692, DOI 10.1090/proc/17017, https://doi.org/10.1090/proc/17017
  9. Jeffrey C. Lagarias, "Euler's constant: Euler's work and modern developments", arXiv:1303.1856 (2013), https://arxiv.org/abs/1303.1856; Bulletin of the American Mathematical Society 50 (2013), 527–628, DOI 10.1090/S0273-0979-2013-01423-X, https://doi.org/10.1090/S0273-0979-2013-01423-X