40 problems

Let the inner prime number distribution law (9) and estimation (25) be those defined in the paper, and let $\zeta(s)$ denote the Riemann zeta function. Conditional usefulness claim…

Let $p_n(m)$ denote the prime-number sequences indexed by $m\in\overline{M}$, let $L$ denote the function used in the paper, and let $c_1$ be a constant independent of $m$. Inner p…

Let $\mathscr{E}_f$ be the class of matrices used in the paper, and let $W_n(A_f)$ denote a web associated with a matrix $A_f$. Extension conjecture. For each matrix…

Let … be the class of logarithmic spirals described in the source. Let $\overline{M}$ denote the index set of the corresponding angle families. Prime Number Spider Web hypothesis.…

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…

Shanks' normalized bias conjecture.

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 $n$ be a positive integer, let $a<n$ satisfy $\operatorname{gcd}(a,n)=1$, and define … … where $\phi$ denotes Euler's totient function. Fiori's upper-bound conjecture. For all…

Let $n$ tend to infinity, let $\lambda\in\mathbb{R}$, and let $\omega(m)=\sum_{p\mid m}1$ denote the number of distinct prime divisors of $m$. Here $p$ denotes a prime and $\phi$ i…

Karatsuba's conjecture. As $x\to\infty$,

Least-prime progression conjecture. One has

Generalized Selberg conjecture A.

Let $count$?

The Legendre constant $1.08366$ is the constant appearing in Legendre's approximation to the prime-counting function. The authors conjecture that this constant satisfies a simple a…

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 $\operatorname{Ri}(x):=\sum_{m=1}^{\infty}\frac{\mu(m)}{m}\operatorname{li}(x^{1/m})$ be the Riemann prime-counting function, and define … where $\mathbf{1}_{\mathcal{P}}(i)$ i…

Let $\operatorname{RM}_c$ be the constrained random model, and let $\pi_{\operatorname{RM}_c}(x)$ denote the associated counting function for a realisation. Let…

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…