25 problems

Let $s\geqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ with smooth metric, and an $s$-step nilsequence is a sequence of the form…

Let $G/\Gamma$ be a $k$-step nilmanifold, let $T_g(x\Gamma)=xg\Gamma$, and let $(F(T_g^n\cdot x))_{n\in\mathbb{N}}$ be a $k$-step nilsequence. The Möbius orthogonality conjecture f…

Let $\mathbb K=\{k_i\}_{i\in\mathbb N}$ be the set of integers produced by the paper's iterative construction, and let … be the set of square-free, or Möbius, numbers. Auil's enume…

Let $S=(q_1,\dots,q_t)$ be a finite sequence of primes and define $\nu_S(n)=(\nu_{q_1}(n),\dots,\nu_{q_t}(n))$. Say that $n$ is $S$-squarefree if $q^2\nmid n$ for every prime…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $g$ be either $g(n)=\mu(n)$ or $g(n)=\mu(n)\log(n)$. Let $\Lambda(n)$ and $\varphi(q)$ denote the…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Eul…

Let $0<\theta<1$ be fixed, let $h\neq0$ be a fixed even integer, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient…

Harper's conjecture. For all $0\leq q\leq 1$ and any fixed $A>0$, for large prime $r$,

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the M…

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the M…

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the M…

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the M…

Let $x\to\fty$. Let $a_0,a_1,\ldots,a_{k-1}$ be a fixed admissible $k$-tuple, and let $b_0<b_1<\cdots<b_{l-1}$ be a fixed subset of small integers. Hardy–Littlewood–Chowla conjectu…

Let $f(n)$ be an arithmetic function, let $\operatorname{idx}(p_n)=n$, let $p_{\operatorname{min}}(n)$ be the prime associated with the least prime factor of $n$, and define…

Veech's conjecture. For any $\eta\in\mathcal{I}_S(\bm \mu)$,

Let $f$ be the Collatz function, let $p>2$ be prime with $p\equiv 3\pmod 4$, and let $\mu$ denote the Möbius function. The Möbius-nonvanishing conjecture. There exists an integer…

Upper-bound conjecture. The values $d_n$ are bounded from above by $0.6040$.

Let $\bm\mu$ denote the Möbius function. For integers $r\geq 0$, shifts $1\leq a_1<\dots<a_r$, and exponents $i_s\in\{1,2\}$ that are not all equal to $2$, consider the logarithmic…

Let $M_\tau(x)$ be the Riesz mean … where $\mu$ is the Möbius function. Let $\rho$ be any non-trivial zero of the Riemann zeta-function, with multiplicity $m(\rho)$, and let…

Let $\pi=\pi_1\oplus\cdots\oplus\pi_n$ be a decomposable permutation with $t$ sequences of equal components … of respective lengths $\alpha_i$, for $1\le i\le t$. Let…

Let $p:\mathbb{N}\to\mathbb{N}_0$ be a polynomial and let $(X,T)$ be a minimal topological dynamical system. For $f\in C(X)$ and $x\in X$, consider the sequence $(f(T^{p(n)}x))$. P…

A minimal topological dynamical system $(X,T)$ has polynomial entropy zero when its sequential topological entropy is zero along all polynomial sequences. Let $(X,T)$ be a minimal…

Let $\mathfrak{A}$ be a unital exact $C^*$-algebra and let $\alpha$ be an endomorphism of $\mathfrak{A}$. The pair $(\mathfrak{A},\alpha)$ is a noncommutative flow. The Möbius func…

Assume $b_k=a_k^2$, with the integers $a_k$ pairwise relatively prime, and define … Let $\Sigma$ denote the parameter from the preceding construction, and let $\beta$ be the probab…

Let $r\geq 1$, let $n_1,\ldots,n_r\in\mathbb{N}$, and let $k_1,\ldots,k_r\in\{1,2\}$, with the $k_i$ not all even. Then … as $N\to\infty$. Chowla's conjecture. The displayed asympt…