17 problems

For an integer $n$ written as $n=\delta+h(q-1)$ by Euclidean division by $q-1$, define … and let $P_n(x)$ be … Let $\zeta_C^*(-n)$ be the first nonzero coefficient in the expansion…

Algebraic independence conjecture for odd zeta values. The numbers $\pi,\zeta(3),\zeta(5),\zeta(7),\zeta(9),\ldots$ are algebraically independent over $\mathbb{Q}$.

For $n\ge 0$, let ${\bf u}_n$ and ${\bf v}_n$ be the sequences defined by the Apéry-type hypergeometric series ${\bf S}_{n,4,2,1,1}(1)={\bf u}_n\zeta(4)-{\bf v}_n$. Define … where…

Chowla–Milnor conjecture. The set $\mathfrak{L}$ is linearly independent over $\mathbb{Q}$. The conjecture is important because its consequences include conditional irrationality r…

Let $q\geq 1$ be an integer, and let $H_k=\sum_{j=1}^k\frac{1}{j}$ denote the $k$th harmonic number. For rational numbers $\alpha_{q,2},\alpha_{q,3},\ldots,\alpha_{q,q+2}$, conside…

Generalized cyclic insertion conjecture. For $a_1,a_2,a_3,b_1,b_2\geq0$,

Cyclic insertion conjecture. For $n\geq0$ and $m_0,\ldots,m_{2n}\geq0$,

For a multi-index $\boldsymbol{k}=(k_1,\ldots,k_d)$, write $\zeta(\boldsymbol{k})$ for its multiple zeta value, and write $\{2\}^m$ for $m$ consecutive entries equal to $2$. For…

Let $(d_n)_{n\geqslant1}$ be the Padovan sequence defined by $d_1=0$, $d_2=1$, $d_3=1$ and $d_n=d_{n-2}+d_{n-3}$ for $n\geqslant4$, and let $(f_n)_{n\geqslant1}$ be the Fibonacci s…

Higher Apéry-limit conjecture. For $d\geq 5$, there is a unique such solution $C^{(d)}(n)$ satisfying

Apéry-limit conjecture. For $d\geq 3$, there is a unique such solution $B^{(d)}(n)$ satisfying

Irrationality-exponent conjecture. The irrationality exponent of the first odd zeta constant satisfies

Logarithmic-integral space conjecture. For every $n\geq 0$,

Logarithmic-integral algebraicity conjecture. For every $n\geq 1$, this value belongs to the algebra over $\mathbb{Q}$ generated by $\pi^2$, $\log 2$, and $\zeta(n)$ for odd $n$. T…

The Riemann zeta function is denoted by $\zeta$, and $\zeta(5)$ is its value at $5$; $1/\zeta(5)$ denotes its reciprocal. Irrationality conjecture. The real numbers $\zeta(5)$ and…

Let $X$ be a Fano manifold and let $\gamma\in H^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X)$ be a homogeneous coprimitive cohomology class of codimension $n$ with respect to $-K_X$…

For positive integers $k,n,s$ with $k\geq n+s$ and $n\geq s\geq 1$, let $X_0^{\star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$, and height…