Apéry-type rational approximations to Euler's constant: a second-order homogeneous recurrence construction, or integer linear forms in 1 and γ of better-than-sub-exponential quality

Euler's constant γ is not known to be rational or irrational. The standard constructive route to irrationality — Apéry's method for ζ(3) — needs integer linear forms q_n γ − p_n whose decay beats the growth of their denominators. All known constructions (Aptekarev et al. 2007; Rivoal 2009; Hessami Pilehrood and Hessami Pilehrood 2011–2013; Wolfs and Van Assche 2025) achieve only sub-exponential quality: errors of order $e^{-c\sqrt{n}}$ or $e^{-cn^{3/4}}$ against denominators growing like $\exp((1+o(1))\,n\log n)$, which no irrationality criterion can use. The problem is to construct Apéry-type approximations to γ in which numerator and denominator both solve one second-order homogeneous linear recurrence with polynomial coefficients, or otherwise to produce integer linear forms in 1 and γ of genuinely better-than-sub-exponential quality, so that an irrationality criterion applies. The most recent literature (through 2026) treats this as open.

Background

Euler's (Euler–Mascheroni) constant is $\gamma=\lim_{n\to\infty}(1+1/2+\cdots+1/n-\log n)\approx 0.5772156\ldots$, equivalently $\gamma=-\int_0^\infty e^{-x}\log x\,dx$; it equals $-\psi(1)$, where $\psi=\Gamma'/\Gamma$ is the digamma function, the logarithmic derivative of the Gamma function. Whether $\gamma$ is irrational is one of the oldest open problems in the arithmetic of constants; it is recorded as open in every account of the subject from Euler's own work to Lagarias's 2013 survey of Euler's constant and modern developments.

A classical and elementary criterion drives the whole subject: if a real number $x=a/b$ is rational, then every rational $p/q\ne x$ satisfies $|x-p/q|\ge 1/(bq)$; consequently, if infinitely many integer pairs $(p_n,q_n)$ with $q_n>0$, $q_n\to\infty$, and $q_nx-p_n\ne0$ have $q_nx-p_n\to0$ — the nonzero integer linear forms in 1 and $x$ tend to zero — then $x$ is irrational. A stronger, power-saving version of the same criterion requires $|q_n x-p_n|\le C\,q_n^{-1-\delta}$ for some constants $C>0$ and $\delta>0$, so that the decay of the integer linear forms beats the denominator growth by a power; this stronger bound is the quantitative target of option (b) below. Refined quantitative versions, such as the linear-independence criterion of Nesterenko, extract from any such imbalance a lower bound on the $\mathbb{Q}$-dimension of the span of the numbers involved.

Apéry's 1979 proof of the irrationality of Apéry's constant $\zeta(3)=\sum_{m\ge1}m^{-3}$ is the archetype. He exhibited integer sequences $(u_n)$, $(v_n)$, both solutions of the single second-order homogeneous linear recurrence $(n+1)^3y_{n+1}-(34n^3+51n^2+27n+5)y_n+n^3y_{n-1}=0$ with coefficients polynomial in $n$, such that $v_n/u_n\to\zeta(3)$ with geometrically decaying error against geometrically growing denominators, and the criterion above applies. By the Poincaré–Perron theory of linear recurrences — solutions of a recurrence with polynomial coefficients grow like products of powers of the characteristic roots, so the ratio of a dominant and a subdominant solution decays geometrically — any pair of integer sequences solving one and the same genuinely second-order homogeneous recurrence with polynomial coefficients and characteristic roots of separated moduli, and converging to a limit, automatically has this geometric balance. A genuinely Apéry-type pair for $\gamma$ is therefore expected to be decisive for irrationality, and its absence is exactly the gap recorded here.

For $\gamma$ the state of the art is far weaker. Aptekarev and collaborators (2007) found the first good rational approximations to $\gamma$, generated by a third-order homogeneous recurrence, with error of order $e^{-2\sqrt{2n}}$ but denominators comparable to $(2n)!$; the linear forms do not tend to zero even after cancelling their large common factor built from factorials and least common multiples. Rivoal (2009) constructed forms from simultaneous Padé approximants (rational simultaneous approximants generalizing continued fractions) to Euler-type integrals, governed by a sixth-order recurrence; the Hessami Pilehroods proved his conjectures on denominators (2011) and reorganized the theory through Meijer G-functions — a standard class of generalized hypergeometric integrals — obtaining approximations whose denominators solve the second-order homogeneous recurrence $q_{n+2}-2(n+2)q_{n+1}+(n+1)^2q_n=0$ while the numerators solve only the corresponding inhomogeneous recurrence, with error of order $e^{-4\sqrt n}$ and denominators $n!\,e^{2\sqrt n}$ up to polynomial factors; the same homogeneous recurrence generates Stieltjes's 1895 continued fraction for the Euler–Gompertz constant $\delta=\int_0^\infty e^{-x}/(1+x)\,dx\approx 0.5963\ldots$. The structural obstruction throughout is factorial/lcm-type coefficient growth: $\log q_n\asymp n\log n$ while $-\log|\gamma-p_n/q_n|=O(n^{3/4})$ at best, so the error is only $q_n^{-o(1)}$ — sub-exponential quality — and no criterion can consume it. The arithmetic nature of $\delta$ is likewise open, though Rivoal (2012), applying Shidlovskii's theorem on values of E-functions (entire functions satisfying linear differential equations with polynomial coefficients), proved that at least one of $\gamma$ and $\delta$ is transcendental, without deciding which.

The frontier of the program (Wolfs and Van Assche 2025, using multiple orthogonal polynomials — polynomials orthogonal with respect to several weights simultaneously, here attached to the exponential integral — and strong asymptotic methods) pushes the error to $\exp(-c\,n^{3/4})$ and proposes families reaching $\exp(-c_p\,n^{(p+2)/(p+3)})$; every known or proposed exponent is $o(n)$, hence still sub-exponential against $\log q_n\asymp n\log n$. The one modification producing exponentially decaying errors inserts a logarithm with an $n$-dependent argument, which destroys the integrality structure the criterion needs. Closing this gap is the problem.

Problem Statement

This is an open problem in the rational-approximation literature for Euler's constant: construct integer sequences $(p_n)_{n\ge0}$ and $(q_n)_{n\ge0}$ with $q_n>0$, $q_n\to\infty$, the two sequences not both identically zero, such that either (a) both sequences are solutions of one and the same genuinely second-order homogeneous linear recurrence with coefficients polynomial in $n$ — that is, a recurrence $A(n)\,y_{n+1}+B(n)\,y_n+C(n)\,y_{n-1}=0$ whose leading coefficient $A(n)$ is not identically zero and whose characteristic roots have separated moduli (their ratio bounded away from $1$ in modulus: $|r_1(n)/r_2(n)|\le\theta<1$ for all large $n$, for the two characteristic roots $r_1(n),r_2(n)$ and some constant $\theta$) — a genuinely Apéry-type construction — and $p_n/q_n\to\gamma$ with $q_n\gamma-p_n\ne0$ for all large $n$ and $q_n\gamma-p_n\to0$; or (b) the integer linear forms in 1 and $\gamma$ satisfy a better-than-sub-exponential quality bound, namely that constants $C>0$ and $\delta>0$ exist with $q_n\gamma-p_n\ne0$ and $|q_n\gamma-p_n|\le C\,q_n^{-1-\delta}$ for all large $n$ (equivalently $|\gamma-p_n/q_n|\le C\,q_n^{-2-\delta}$). By the classical criterion recalled in the Background, (a) with its forced geometric decay, or (b) directly, implies that $\gamma$ is irrational. Every known construction gives sub-exponential quality only — $|\gamma-p_n/q_n|$ of size $q_n^{-o(1)}$, with errors $e^{-c\sqrt n}$ (Aptekarev et al. 2007; Hessami Pilehrood 2013) or $e^{-cn^{3/4}}$ (Wolfs–Van Assche 2025) against $\log q_n\asymp n\log n$.

References

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