9 problems

Rationality conjecture. The sum is a rational number if and only if $k=x=1$. This conjecture concerns when the $p$-adic sum of the factorial power series can be rational; the suppl…

Rational-function series conjecture. This number is either rational or transcendental, and it is never a Liouville number. Moreover, if it is rational, then the series is telescopi…

Let $H_n=\frac1{1}+\frac1{2}+\fdots+\frac1n$ denote the $n$th harmonic number, with $H_0=0$, and let $G$ and $K$ denote the constants appearing on the right-hand sides of the ident…

Heuristic threshold conjecture. The series converges for $\theta>\frac{1}{2}$ and diverges for $\theta\leq\frac{1}{2}$. This stronger conjecture is motivated by probabilistic heuri…

The special-value conjecture. The following identity should hold:

Stolarsky's conjecture. The series cannot be rational for every positive integer $t$.

Let $H:\mathbb{Z}\rightarrow\mathbb{Z}$ be a $p$-Hydra map, where $p$ is a finite prime, and suppose its numen $\chi_H:\mathbb{N}_0\rightarrow\mathbb{Q}$ extends to a function…

Double-series formula. The quantity satisfies

Let $R_M$ be the remainder integral in, and suppose its integrand has the oscillatory nature described in the surrounding analysis. Oscillatory remainder estimate conjecture. A mor…