49 problems

Let $\mathcal A=\{a_1,\ldots,a_s\}$ be an admissible set of distinct natural numbers, meaning that the associated linear forms satisfy the usual local non-obstruction condition. Ch…

Let $x$ be a parameter, let $u 1$, and let $\mathcal{P}$ be a subset of the primes at most $x$. Define … The integers are sieved by the primes in…

Maier–Pomerance conjecture. Maier and Pomerance conjectured that

Let $f(t)$ be an irreducible polynomial of degree at least $4$. For $N>0$, consider integers $t\in[N,2N]$ for which $f(t)$ is divisible by the square of a prime larger than…

Let $\mathcal{B}$ be the set defining the $\mathcal{B}$-free numbers. Erdős' short-interval conjecture. For every $\varepsilon>0$, there exists $N_{\mathcal{B},\varepsilon}$ such t…

Let $\Delta_Q(x)$ denote the pairing defect function introduced in the paper for the residue-pairing framework associated with the residue-pairing bound $B_Q(x)$. Pairing-defect pa…

Let $A$ be a set whose elements are all products of an odd number of primes, or all products of an even number of primes. Parity problem. Without injecting additional ingredients,…

For a squarefree integer $Q$ with $6\mid Q$, let $U_Q$ be the set of residues $r\bmod Q$ such that both $r$ and $r+2$ are coprime to $Q$. Define the residue-pairing bound … where ……

For integers $k\geq 2$ and $q\geq 1$, define the error term $E_{k,q}(x)$ by … where … Here $\varphi(q)$ is Euler's totient function, $\theta(q)$ is the number of positive squarefre…

Let $C_1$ denote the constant governing the sieve estimates used to bound the lower density of integers representable as a prime plus a power of $2$. Elliott–Halberstam-related con…

Let $p_k$ be a prime, let $\mathcal{G}(p_k^{\#})$ be the corresponding Eratosthenes-sieve cycle, and let $\Delta H(p_k)=[p_k^2,p_{k+1}^2]$ be its interval of survival. The populati…

Prime-increment conjecture. For every $n\geq 2$, we have $b_n\in\{1\}\cup\mathbb{P}$, where $\mathbb{P}$ denotes the set of prime numbers.

Let $S([1,N] \cap \mathbb{Z},z;\Omega)$ be the set of integers $n\leq N$ such that, for every prime $p<z$, one has $n\pmod p\notin\Omega(p)$, where $\Omega(p)$ is a set of residue…

Let $\chi$ be the chaotic map used in the EMCHS model, let $y_0\in[0,1]$, and let $\rho(y)$ denote the proposed invariant density. For a suitable class of functions $g$, bounded-ch…

Let $p_n$ denote the $n$th prime, and let $\delta$ and $\epsilon$ be parameters in the ranges considered by the Enhanced Multidimensional Chaotic Heuristic Sieve (EMCHS), including…

Let $\widetilde{M}(k)$ denote the parameter in Theorem. Conjecture on $\widetilde{M}(k)$. One can take $\widetilde{M}(k)=k$ for all $k\geq 1$ in Theorem. This would improve the pap…

The paper studies sequences $w_n$ supported on integers with no divisor in $((x/2)^\theta,x^{\theta+\nu}]$ and satisfying the Type I and Type II conditions referenced in the constr…

Let $A\subseteq\mathbb{N}$ be admissible if there is no prime $p$ such that $A$ contains at least one element in every residue class modulo $p$. Erdős and Graham's conjecture. Ther…

Limiting coprimality conjecture. The constant $c_{a,b}$ satisfies

Let $k$ be a positive integer, and define … Here $(k,2)$ denotes the greatest common divisor of $k$ and $2$. The conjecture for $C_k$. The limit exists and satisfies … This follows…

Let $X$, $y$ and $H$ be parameters satisfying the conditions denoted by … , and let $\alpha_y(n)$ be the indicator of integers without prime factors at most $y$, with mean density…

For $k\geq0$, define … where $\Lambda$ is the von Mangoldt function and $\mathfrak{S}(k)$ is the singular series. The weakened prime-pair upper-bound conjecture. Given a fixed cons…

For an integer $m$, call a prime factor of $m$ small when its size is $O(\log n)$ in the surrounding construction; the claim concerns consecutive integers below $X$ that each conta…

Let $\delta_2$ be the constant appearing in lower-bound sieve results for almost primes: if a set has level of distribution $x^{\vartheta-\varepsilon}$ with…

Let $W_y$ be the minimum of $|[0,y]\cap\mathcal{S}_{(y/\log y)^{1/2}}|$ over all choices of residue classes modulo primes up to $(y/\log y)^{1/2}$, and define … Let…