Is Euler's constant a value of an arithmetic Gevrey series? Membership of $\gamma$ in the rings $\mathbf{E}$, $\mathbf{G}$ and $\mathbf{D}$

Euler's constant $\gamma=-\Gamma'(1)\approx 0.5772$ is not known to be irrational. Siegel's theory of E-functions and G-functions, and its modern extension by André's E-operators, yields the three rings $\mathbf{E}$, $\mathbf{G}$ and $\mathbf{D}$ of complex numbers obtained as values, at algebraic points, of arithmetic Gevrey series of order $-1$, $0$ and $1$ respectively; these rings capture essentially all constants reachable by the Siegel–Shidlovskii transcendence machinery. The problem is to decide, unconditionally and for each of the three rings, whether $\gamma$ belongs to it. All three membership questions are open: Fischler and Rivoal have shown conditional results (their Conjectures 1 and 2 would imply $\gamma\notin\mathbf{E}$) and experts expect $\gamma$ to lie in none of the three rings, but no unconditional result exists in either direction, and for $\mathbf{G}$ and $\mathbf{D}$ not even a conjectural proof route is known.

Background

A power series $\sum_{n\ge 0} a_n z^n/n!\in\overline{\mathbb{Q}}[[z]]$ with algebraic coefficients is an E-function (in Siegel's sense) if it satisfies a linear differential equation over $\overline{\mathbb{Q}}(z)$ — such series are called holonomic or D-finite — and its coefficients have at most geometric growth under every embedding of the coefficient field, together with at most geometric growth of the least common denominators of $a_0,\dots,a_n$; the archetypal examples are $e^z$, Bessel functions and confluent hypergeometric series with rational parameters. Replacing $a_n/n!$ by $a_n$ and requiring the series to have a positive radius of convergence gives G-functions, such as $\log(1+z)$ and algebraic functions; their values at algebraic points include logarithms of algebraic numbers. Siegel's 1929 memoir and the Siegel–Shidlovskii theorem (values of E-functions at algebraic points are as algebraically independent as the functions themselves) make these classes the backbone of explicit transcendence theory. In 2000 Y. André introduced E-operators: linear differential operators whose coefficient functions are G-functions; every E-function satisfies such an operator, and André constructed local bases of solutions and purity theorems for them. Fischler and Rivoal developed the arithmetic theory of these operators (connection constants at finite distance and Stokes constants at infinity — the constants governing the asymptotic expansion of solutions along a direction at $\infty$) and found that Euler's constant $\gamma=-\Gamma'(1)$, the Euler–Mascheroni constant, enters this theory in an unavoidable but indirect way: in all Stokes data it appears only in the combination $\gamma-\log(1/z)$. This led them to frame the present question in terms of arithmetic Gevrey series, i.e. holonomic series whose coefficients grow like $n!^s$ for a rational Gevrey order $s$: order $s=-1$ gives E-functions, order $s=0$ gives convergent G-functions, and order $s=1$ gives the divergent series that occur as formal solutions of E-operators at infinity (also called Э-functions); renormalizing such divergent Gevrey-1 series by summation along a non-singular direction produces numbers. Writing $\mathbf{E}$, $\mathbf{G}$ and $\mathbf{D}$ for the sets of values at algebraic points obtained from the three orders respectively (each is a ring containing $\overline{\mathbb{Q}}$), Fischler–Rivoal proved that $\mathbf{G}\subseteq\overline{\mathbb{Q}}[\mathbf{E}\cup\mathbf{D}]$ and that $\mathbf{D}$ contains the Gompertz constant $\delta=\int_0^\infty e^{-t}/(1+t)\,dt$, while $\gamma\in\mathbf{E}+e^{-1}\mathbf{D}$ unconditionally. Euler's constant has hundreds of analytic representations (integrals, series, limits, continued fractions), yet, as Rivoal documented in a 2017 survey note, none of them exhibits $\gamma$ as a value at an algebraic point of an E-function, a G-function, or a Mahler function (a function $f$ satisfying a self-functional equation relating $f(z)$ to $f(z^d)$ for an integer $d\ge 2$, such as $\sum_{n\ge 0}z^{2^n}$), and it is plausible that no such representation exists — so the arithmetic nature of $\gamma$ may lie outside the reach of the classical Siegel–Shidlovskii theorem on E-function values and its extensions by Chudnovsky and Nishioka (which extend such algebraic-independence machinery to G-function values and related settings). Whether $\gamma$ belongs to $\mathbf{E}$, to $\mathbf{G}$ or to $\mathbf{D}$ is therefore a precise and delicate structural question about the most famous constant whose irrationality is still unknown; since $\overline{\mathbb{Q}}$ is contained in all three rings, membership would not by itself prove anything about irrationality, while non-membership would be the first unconditional structural-arithmetic fact ever established about $\gamma$.

Problem Statement

Decide, by unconditional proof, for each of the three rings of arithmetic Gevrey-series values defined by Fischler–Rivoal — $\mathbf{E}$ (values at algebraic points of E-functions, arithmetic Gevrey series of order $-1$), $\mathbf{G}$ (values at algebraic points of the analytic continuations of G-functions, order $0$), and $\mathbf{D}$ (values at algebraic points of the renormalizations, along non-singular directions, of divergent Gevrey-1 series solving E-operators at infinity, order $1$) — whether Euler's constant $\gamma=-\Gamma'(1)$ belongs to that ring. A complete answer determines the membership triple $(\gamma\in\mathbf{E}?,\,\gamma\in\mathbf{G}?,\,\gamma\in\mathbf{D}?)$. An unconditional proof that $\gamma\notin\mathbf{E}$ and an unconditional proof that $\gamma\notin\mathbf{G}$ or $\gamma\notin\mathbf{D}$, as well as an unconditional proof of membership in any one of the three rings (by an explicit arithmetic Gevrey series and algebraic evaluation point), each resolve a component of the problem as stated; a full solution resolves all three. The rings are to be understood exactly as defined in the cited literature, with no restriction on the differential equation, the coefficient field (a number field), or the algebraic evaluation point.

References

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