73 problems

Heil–Ramanathan–Topiwala conjecture. For every nonzero $g\in L^2(\mathbb{R})$ and every finite set of distinct points $\Lambda=\{(a_k,b_k)\}_{k=1}^N\subset\mathbb{R}^2$, the system…

Let $\zeta(s)$ be the Riemann zeta function, and let $\gamma>0$ range over the positive imaginary ordinates of its nontrivial zeros $\rho=\beta+i\gamma$. Linear Independence Conjec…

Linear Independence Conjecture. This set is linearly independent over $\mathbb{Q}$. The conjecture is used in the paper's large-deviation arguments for the summatory functions unde…

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 the positive ordinates of the zeros of all principal automorphic $L$-functions be counted with multiplicity. Grand Simplicity hypothesis. This multi-set of positive ordinates i…

Number-field linear independence conjecture. The numbers

Let $N$ be a positive integer, and let $\Phi^{(N)}$ be the finite Fourier transform with matrix entries … For each nonnegative integer $n$, define … where $H_n$ is the Hermite poly…

Assume the nontrivial zeros of the zeta function are written as $\rho_n=1/2+i\gamma_n$. LI conjecture. The imaginary parts $\gamma_n$ of the nontrivial zeros are linearly independe…

Linear-independence conjecture. If $f\ne 0$, then $S(f,\Lambda)$ is linearly independent over $\mathbb{C}$. This problem is motivated by Gabor analysis; the source gives no evidenc…

Define the multiple zeta values by … for $s_1\geq 2$ and $s_2,\ldots,s_p\geq 1$. Selected-value independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, $\zeta(2,3)$…

Define the multiple zeta values by … for $s_1\geq 2$ and $s_2,\ldots,s_p\geq 1$. Low-depth independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, and…

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…

Witness-elimination conjecture. With

Linear-independence conjecture. Fix a rectangle $R$. The products

Let $q\in\mathbf{Z}_{\geqslant 1}$, let $A>1$, and let $\mathscr X_q$ be a set of Dirichlet characters modulo $q$. Enumerate in non-decreasing order, with multiplicities, the posit…

For $q\geqslant 3$, consider the multiset of positive imaginary parts of the zeros of all Dirichlet $L$-functions associated with characters modulo $q$. The linear independence con…

Generalized Linear Independence Conjecture. For any fixed $q\geq 3$ and $\kappa\geq 1$, this set is linearly independent over $\mathbb{Q}$. The conjecture generalizes linear indepe…

Let the combinatorial (bi-)multiple Eisenstein series be the $q$-series in ${\mathbb Q}\llbracket q\rrbracket$ introduced by Bachmann and Burmester through Bachmann's stuffle regul…

KTB19's linear-independence conjecture. There exists $\tilde{r}=\tilde{r}(d,k)$ such that $p_1^r,\ldots,p_k^r$ are linearly independent whenever $r>\tilde{r}$.

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

Weak mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

Canonical splitting conjecture. There is a canonical direct-sum decomposition

Mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

Beilinson's linear independence conjecture. The values $L^{(N)'}(-1,x)$, for $x\in U$, are linearly independent over ${\mathbb Q}$. This is presented as a consequence of the Beilin…

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…