19 problems

Let $r,D\geq 1$ be integers, and let $\Delta_{r,D}$ denote the determinant of the linear system obtained from the Bernoulli-polynomial identities for the coefficients of the restri…

Bernoulli determinant conjecture. The first determinant equals

Bernoulli-moment sign conjectures. For every $k\in{\mathbb N}$, both the weak form

For $k\geq1$, define … and let $\mathcal{S}_k$ be the corresponding computable subsets, with … … Second- and third-derivative completeness conjecture. One has … The paper proves fi…

Define … Radical characterization conjecture. One has … The surrounding results show that $\overline{\mathcal{R}}$ is finite and constrain its elements; the conjecture remains unre…

The Bernoulli polynomials $\mathbf{B}_n(x)$ are defined by … and let … Finiteness and completeness conjecture. The set $\overline{\mathcal{S}}$ is finite and … This conjecture asse…

For each positive integer $n$, let $b_n(u)$ denote the factorially normalized Bernoulli polynomial of degree $n$, and consider the polynomial probability density $1-2^n b_n(u)$ on…

Let $b_n$ be the factorially normalized Bernoulli polynomial of degree $n$, and let $\delta_{k:2n}$ denote the difference between the uniform distribution on $\{1,\dots,2n\}$ and t…

Positivity conjecture. For all $x\in\mathbb{R}$ and $n\in\mathbb{N}$,

Let $D\geq 2$, and let $\bar F_{2,D}(x_1,\ldots,x_D)$ be the determinant polynomial defined in the paper for $r=2$. Quadratic-index factorization conjecture. There is a rational co…

Let $r\geq 1$, and let $\bar F_{r,2}(x_1,x_2)$ be the determinant polynomial defined in the paper for $D=2$. Factorization conjecture. One has … The formula was supported by comput…

Let $r,D\geq 1$, and let $\Delta_{r,D}$ denote the determinant defined in the paper from Bernoulli polynomials. Nonvanishing conjecture. The determinant is nonzero: … This conjectu…

Let $z$ and $N$ be integers with $\gcd(z,N)=1$. Let $B_1$ and $B_2$ denote the first and second Bernoulli polynomials, respectively, and interpret $z(k-\ell)\bmod N$ as the residue…

For a positive integer $n$, let $s_p(n)$ denote the sum of the base-$p$ digits of $n$, and define … Let $P(m)$ be the largest prime factor of $m$. Kellner's conjecture. For $n>192$…

Let $a,r\in\mathbb{Z}$ with $a\geq0$ and $r\geq1$, let $B_r(x)$ denote the $r$-th Bernoulli polynomial, and let $\{x\}$ denote the fractional part of $x$. Bernoulli polynomial sum…

Let $k\geq 2$ be even, and let $B_k(x)$ denote the $k$-th Bernoulli polynomial. The irreducibility conjecture. The polynomial $B_k(x)$ is irreducible over the rationals for every e…

For $j\in\mathbb N_0$, let $B_{2j+1}^\sim(z)$ be the Hilbert-transformed Bernoulli function, let $B_m(z)$ denote the Bernoulli polynomials, let $\eta(s)$ be the Dirichlet eta funct…

The Bernoulli functions $B_\alpha(z)$ and their Hilbert transforms $B_\alpha^\sim(z)$ are considered for $\alpha\in\mathbb C$ and $z\in\mathbb C\setminus\mathbb R_0^-$. Let…

Let $B_n(x)$ denote the Bernoulli polynomial of degree $n$, and let $p>3$ be prime. Sun's infinitude conjecture. There are infinitely many primes $p>3$ such that … The paper report…