33 problems

Lang–Rohrlich conjecture. The value $\Gamma(z)$ is algebraic if and only if

Let $n$ be a positive integer and consider the Gamma values … The reflection formula and the Gauss multiplication formula give relations among these values over…

AVV conjecture. The function $H$ is increasing from $(1,\infty)$ into $(1/100,1/30)$. This conjecture is attributed to the authors' earlier work and is motivated by Ramanujan's rec…

Let $R_n=\sum_{k=1}^n\frac{1}{k}-\log\left(n+\frac{1}{2}\right)$, $h(n)=R_n-\gamma$, and $H(n)=n^2h(n)$ for $n\geqslant1$, where $\gamma$ is the Euler–Mascheroni constant. Extendin…

Let $b$ and the integers $m_a$, indexed by rational numbers $a$, be integers, and consider a multiplicative relation … Here $\Gamma$ is Euler's Gamma function, and the product is u…

The four numbers … are considered, where $\Gamma$ is Euler's Gamma function. Algebraic-independence conjecture. At least three of these four numbers are algebraically independent.…

Real Manin–Mumford conjecture for the Gamma function. Conditional on Conjecture, if $W$ is an irreducible real algebraic curve in $\mathbb R^2$ containing infinitely many Gamma-spe…

Let $\widehat{m}_{1/2}(x,\alpha)$ denote the logarithmic error term for the shifted interpolation parameter $\alpha$, let $\widehat{D}_{1/2}$ be its stated domain, and let…

The conjectured optimal logarithmic interpolation bound. For all $(x,\alpha)\in\widehat{D}$,

Let $n$ be a positive integer. For coordinate projections $\mathrm{Pr}_{\mathbf{i}}$ onto any selected $\ell$ pairs of coordinates, call an algebraic set…

Let $f_1,\ldots,f_n$ be holomorphic functions satisfying, for every distinct $i,j\in\{1,\ldots,n\}$, $f_i\notin\mathbb{Q}f_j+\mathbb{C}$. Write $\mathbf{f}=(f_1,\ldots,f_n)$ and…

Let $\tilde{\Gamma}(x)$ denote the principal branch of the inverse of the gamma function, so that $\tilde{\Gamma}(x)=y$ when $Gamma(y)=x$. Let $alpha=1.4616ldots$ be the unique p…

Let $R_n(x)$, for $n\geq 0$, be the remainders of the asymptotic expansion of $\log\Gamma(x)$, defined by … For a completely monotonic function $f$ on $(0,+\infty)$, its completely…

Let $\pi_n$ denote the hyperelliptic constant associated with the index $n$. Let $n\in\mathbb{N}$ be composite and not a prime power, so that $n\neq p^m$ for $p\in\mathbb{P}$ and…

Let $C$ be the genus-two hyperelliptic curve over $\overline{\mathbf{Q}}$ given by $y^2=1-x^5$. Its periods can be expressed using $\pi$, $\Gamma(1/5)$, and $\Gamma(2/5)$. Period c…

Asymptotic-expansion conjecture. Numerical exploration suggests that this formal expansion is an asymptotic expansion of the left-hand side as a function of $k$ when $k\to\infty$.

Gun–Ram Murty–Rath conjecture. The dimension of $\overline{V_{\Gamma}(q)}$ is

Gamma-quotient monotonicity conjecture. The quotient decreases as $c$ increases.

Let $a_1, a_2, \ldots, a_r$ be rational numbers that are not in $\{0, -1, -2, \ldots\}$. Let $D$ be a common denominator of the $a_i$'s. Rohrlich's conjecture. The product … is an…

Linear independence conjecture. The values $\Gamma(s)$, for $s\in\mathbb{Q}$ with $0<s\leq 1$, are $\mathbb{G}[\gamma]$-linearly independent; consequently, $\mathbf{S}$ is the free…

The monotonicity conjecture. For every $\tau\geq0$, the function $g_\tau$ is strictly increasing on $(0,\infty)$. This conjecture is one of the monotonicity problems arising from r…

The Ramanujan remainder conjecture. The function $h$ is increasing from $(1,\infty)$ into $(1/100,1/30)$. This conjecture seeks a monotonic formulation of the sharp constants in Ra…

For $x>0$, define … A function is completely monotonic on an interval if $(-1)^n f^{(n)}(x)\ge 0$ for every nonnegative integer $n$ and every point of the interval. Qi–Guo's conjec…

Under the hypotheses and notation of Theorem, let $g(\mu,x)$ denote the reciprocal-gamma series used there, with $x\geq0$. For $\alpha,\beta>0$ and parameters satisfying…

Let $f_k\geq0$ be a sequence, and consider the reciprocal-gamma series … Under the coefficient hypotheses considered in the paper, ordinary log-concavity conjecture. The function…