50 problems

Let $\zeta(s)$ denote the Riemann zeta function, and let $\pi$ be the usual circular constant. The odd positive integer values $\zeta(3),\zeta(5),\ldots$ are the values of the zeta…

Let $A^{(s)}(n)=\sum_{k=0}^n\binom{n}{k}^s\binom{n+k}{k}^s$ be the generalized Franel numbers, and let $m$ be a positive integer. An Apéry limit is a limit of the quotient of two s…

Bailey–Borwein–Girgensohn conjecture. When $p+q>7$ and $p+q$ is even, the double linear sums $S_{p,q}$ are not reducible to zeta values.

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 $E\geq 2$ and $n\geq 0$ be integers, and define … where … Also let $\operatorname{d}_n=\operatorname{lcm}\{1,2,\ldots,n\}$. Vasilyev's conjecture. For all integers $E\geq 2$ an…

Folklore conjecture. The numbers $\pi$, $\zeta(3),\zeta(5),\zeta(7),\ldots$ are algebraically independent over the rationals.

Let $m,k,n$ be positive integers with $k\geqslant n$, let $f(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$ be a symmetric polynomial, and let … For the case $m=1$, set…

Let $\zeta(s)=\sum_{k=1}^{\infty}k^{-s}$ be the Riemann zeta function, and let $\mathbb{N}$ denote the positive integers. For each $n\in\mathbb{N}$, consider the ratio…

For integers $s\geq 2$, let $\zeta(s)$ denote the Riemann zeta function at $s$. Depth-one linear-independence conjecture. The numbers $1,\zeta(2),\zeta(3),\zeta(4),\ldots$ are line…

For positive integers $s\ge 2$, define the Riemann zeta function by … The values at positive even integers are known to be rational multiples of powers of $\pi$, so the conjecture…

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…

Let $n$, $A$, $B$, and $C$ satisfy the conditions imposed above, and let ${\bf p}_{0,C,n}(z)$ and ${\bf p}_{l,n}(z)$ be the coefficient functions occurring in the specialized hyper…

Let $a\geq 3$ be an even integer and let $1\leq r<a/2$. For $n\geq 1$, let $d_n$ be the least common multiple of the integers from $1$ to $n$. Let ${\bf R}_n(k)$ and ${\bf S}_n(z)$…

Let $c0$ be the circle constant and let $b6(s)=a3_{n=1}^{\infty}n^{-s}$ denote the Riemann zeta function for integers $s\geq 2$. Algebraic independence conjecture for odd zeta valu…

Nonvanishing conjecture for the zeta(3)-square class.

Linear independence conjecture. The number $\pi$ does not belong to the rational span of the sequence $(H_p)_{p\geq 1}$:

Let $n\in\mathbb{N}^*$, and consider the integral … For each $n$, let $C_{p,n}$ denote rational coefficients indexed by $1\leq p\leq n$. Integral representation conjecture. For eve…

For each integer $n\geq 1$, the Riemann zeta value $\zeta(2n+1)$ is conjectured to be transcendental. Odd zeta-value transcendence conjecture. … Even positive zeta values are known…

Grothendieck's period conjecture. This period map is injective. A consequence is that the $\mathbf{Q}$-algebra generated by $\zeta(2)$ and the odd values $\zeta(2k+1)$ for $k\geq1$…

Euler's parity conjecture. All $S_{p;q}$ with $p+q$ odd can be expressed as a $\mathbb{Q}$-coefficient combination of zeta values.

Let … and let $\operatorname{span}_{\mathbb{Q}}(\mathcal{A})$ denote the vector space of rational linear combinations of these zeta values. Non-membership conjecture. … This conjec…

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…

Let a Calabi–Yau family have a point of maximal unipotent monodromy at $t=0$, and let $calpha_j$ denote the constants appearing in the Frobenius structure at that point. Candelas–d…

Let $\zeta(s)$ denote the Riemann zeta function, and let $\mathcal{L}_{\beta,M}$ be the set of quotients of derivatives of twisted $L$-functions defined in the paper for the parame…

A Lambert-series evaluation of an odd zeta value has an integer merit $a$ when the asymptotic ratio of successive terms is $\exp(-\pi\sqrt{a})$. The paper considers evaluations of…