33 problems

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 $p_k$ 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 $\chi$ 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…

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 $A\subseteq\mathbb{N}$ 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…

Iwaniec's conjecture. The limit exists and equals

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…

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…

Gaussian Primes Conjecture. One has

Let $a,b$ be positive integers with $\gcd(a,b)=1$, let $x$ and $y$ be positive real parameters, and let $p(n)$ denote the smallest prime divisor of the integer $n$. Sieve bound con…

Let $P(x)\in\mathbb{Z}[x]$ be square-free. The square-free sieve conjecture. The set of integers $n$ for which $P(n)$ is divisible by the square of a prime larger than $\sqrt n$ ha…

Green's inverse conjecture for the large sieve. Then $|A|\ll X^\epsilon$ unless $A$ is contained, apart from a set of size $X^\epsilon$, in the set of values of a quadratic polynom…

Let $X\subset [N]$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue classes. F…

Let $X\subset\mathbb{Z}$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue clas…

Inverse sieve conjecture. At least one of the following holds: $|X|\ll_{\epsilon}N^{\epsilon}$, or there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $d\in[2,C]$ and height…

Let $X\subset [N]$ be a subset. Suppose that, for every prime $p$, the reduction $X\pmod p$ occupies at most $0.9p$ residue classes. Inverse sieve conjecture, rough form. Either…

Larger sieve sharpness conjecture. Under these hypotheses,

Let $\mathcal P$ be a bounded primitive integral Apollonian packing. Define … Here $N_{\mathcal P}(T)$ denotes the relevant circle-counting function. Fuchs–Sanden conjecture. As…