18 problems

A strong form of Goldbach's conjecture. For every even integer $n\geq 6$,

Let $\Gamma_{\mathbb P}^{\mathrm{cyc}}$ be the prime-cyclotomic gamma function. A prime Hankel representation should consist of an oriented contour…

Let $n$ be a positive integer, let $a<n$ satisfy $\operatorname{gcd}(a,n)=1$, and define … … where $\phi$ denotes Euler's totient function. Fiori's explicit two-sided conjecture. F…

Fiori's lower-bound conjecture. For $0<\epsilon<1$, as $y\to\infty$,

Let $\epsilon>0$, and let $E_\epsilon(x,y)$ denote the counting function introduced in the paper for the relevant large outliers of $P(n)$; the supplied context does not reproduce…

Let $n$ be a positive integer, let $a<n$ satisfy $\operatorname{gcd}(a,n)=1$, and define … … where $\phi$ denotes Euler's totient function. Wagstaff's conjecture. As $n\to\infty$,…

Let $p_n$ be the $n$th prime, let $\mathcal P$ denote the set of partitions, and let $\widehat{N}$ be the supernorm map. A partition has no $1$'s when none of its parts equals $1$,…

Let $f(n)$ be an arithmetic function, let $p_n$ denote the $n$th prime, and let $\operatorname{idx}(p_n)=n$. Suppose that the Cesàro mean … exists. Prime-indexed Abelian limit conj…

Unboundedness conjecture. For every $N\geq N_0$, where $N_0>0$ is sufficiently large, the supremum of these partial sums satisfies

Let $\operatorname{RM}$ be the random model, with a realisation determined by the shifts $a$ and factors $\rho_k(n+a)$, and let $\pi_{\operatorname{RM}}(x)$ be its counting functio…

Let ${\operatorname{RM}}_c$ denote the constrained ${\operatorname{RM}}$ described in the source, and let $\pi_{\operatorname{RM}_c}(x)$ be its prime-counting function. Let…

Let $\psi(x)=\sum_{p^k\leq x}\log p^k$ be the weighted prime-counting function. Assume the Riemann hypothesis, and let the nontrivial zeros of the zeta function be written as…

Let $<\varepsilon<1$ and let $\Phi(x)=x^{\varepsilon}$, where $\Phi(x)$ determines the intervals $I(x,\Phi)=[x-\Phi(x),x+\Phi(x)]$. Write $\pi(I(x,\Phi))$ for the number of prime i…

Let $a,q$ be positive integers such that $(a(a+2),q)=1$. Let $\Pi_2$ denote the twin-prime constant, let $\varphi_2(q)$ denote the corresponding arithmetic-progression factor, and…

Let $\mathbb{P}$ be the set of primes, and let $r_2(n;\mathbb{P})$ denote the number of representations of $n$ as a sum of two primes; let $\varphi$ be Euler's totient function. Co…

For a prime $p$, let $H(p)$ be the height of its Pratt tree: the tree with root $p$, whose children are the prime factors of $p-1$, constructed recursively, stopping at $p=2$. Let…

Schur-type conjecture. For every $l\geq 3$, there exist monochromatic primes $p_0,p_1,p_2,\ldots,p_l$ such that $p_1,\ldots,p_l$ form an arithmetic progression with common differen…

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let “the growth assumptions of Theorem A (or Conjecture A)” refer to the hyp…