16 problems

Bessel common-zero conjecture. If $f$ and $J_\alpha$ share a common zero in $\mathbb C^*$, then

Irreducible common-zero conjecture. If $h_1$ and $h_2$ have at least one common zero in $\mathbb C$, then

Bezout non-principality conjecture. The ideal $(f_1,f_2)$ is not principal in general; equivalently, there are no Bezout relations in $\mathcal E$ in general.

Ritt-type factorization conjecture. For every non-zero $E$-function $f$, there exist a unit $u\in\mathcal E^\times$, simple normalized $E$-functions $s_1,\ldots,s_p$ with pairwise…

Simple-factor obstruction conjecture. The failure of factoriality in $\mathcal E$ occurs only with simple functions.

Bessel irreducibility conjecture. The $E$-function $\Gamma(\alpha+1)x^{-\alpha}J_\alpha(x)$ is irreducible in the ring of $E$-functions.

Jossen's factorization conjecture. Every non-zero $E$-function $f$ can be written as a finite product of powers of $E$-functions with simple zeros, with distinct factors having no…

Multiplicity factorization conjecture. Multiple zeros of $f$ always occur for a trivial reason: the zeros of multiplicity $j$ are exactly the zeros of an $E$-function $g_j$ such th…

Jossen's conjecture. (i) If $f$ and $g$ are two $E$-functions such that $f/g$ is entire, then $f/g$ is an $E$-function. (ii) If two $E$-functions $f$ and $g$ share at least one com…

Let ${\bf E}$ and ${\bf M}$ denote the sets of values under consideration of $E$-functions and $M$-functions, respectively. A complex number is an $S$-number when its Mahler expone…

A mixed function is a formal series … where $F$ is an $E$-function and $\mathfrak{f}$ is an arithmetic Gevrey function. For a non-anti-Stokes direction $\theta$, write…

Let ${\bf E}$ and ${\bf D}$ denote the sets of values associated with $E$-functions and arithmetic Gevrey functions, respectively, and let $\overline{\mathbb Q}$ be the field of al…

Let ${\bf E}$, ${\bf G}$, and ${\bf D}$ denote the sets of values under consideration in the paper, with $\overline{\mathbb Q}$ the field of algebraic numbers. Intersection conject…

Let $f$ be an $E$-function and let $z_0\in\overline{\mathbb Q}$. For any $\varepsilon>0$, there exists $c>0$ such that for every $(p,q)\in\mathbb Z\times\mathbb N$ with $q\neq 0$,…

An $E$-function in Siegel's sense is a power series with algebraic coefficients satisfying Siegel's growth conditions: in the coefficient and denominator bounds defining an $E$-fun…

No-E-approximation conjecture. Euler's constant $\gamma$ does not have $E$-approximations.