29 problems

Let $X$ be large, let $\delta$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $X$ be large, let $h$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

For a real number $\theta>0$ and sufficiently large $x$, the interval $[x,x+x^\theta]$ contains a prime number. The short-interval prime conjecture. Given $\theta>0$, for all suffi…

Square-prime short-interval conjecture. For all $x \geqslant 2$,

Let $\pi(t)$ denote the number of primes at most $t$. For a real parameter $\lambda=(\log x)^{o(1)}$ and an integer $k\ll (\log_2 x)^2$, count integers $n\leqslant x$ according to…

Let $\lambda:\mathbb{N}\to\{+1,-1\}$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ counts prime factors with multiplicity. Let $H=H(X)$ be…

Let $\omega(n)$ denote the number of distinct prime factors and define … where $\log_2 n=\log\log n$ and $\log_3 n=\log\log\log n$. Short-interval prime-factor conjecture. For some…

For $x>0$, define … where $\log_2 n=\log\log n$. Short-interval conjecture. For $\varepsilon>0$, there is a constant $C$ such that, for sufficiently large $x$, … and … This conject…

Let $n\geq 1$ be an integer, let $\mathcal{D}_n$ be the set of positive divisors of $n$, and define … Fix real numbers $\theta$ and $\epsilon$ satisfying $0<\epsilon<\theta<1$. Sho…

Variance conjecture.

Extended tail-bound conjecture. As $x\to\infty$,

Poisson conjecture. As $x\to\infty$,

Let $k\ge 2$ be an integer, let $x\ge 1$ be real, and let $Q_k(t)$ denote the number of $k$-full numbers at most $t$. A number is $k$-full if every exponent in its prime factorizat…

Variance asymptotic conjecture. The variance of the count of squarefull numbers in these intervals satisfies

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…

Let $U_n$ be uniformly distributed on $\{n+1,\ldots,2n\}$, let $d_k(N)$ be the $k$-th divisor function, and set $\Delta_k(U_n)=d_k(U_n)-\mathbb{E}(d_k(U_n))$ and…

For $\alpha>0$ and $k\in\mathbb N$, define $\mathcal C_k(\alpha)$ as the normalized limiting variance of the $k$-fold divisor function in short intervals, and let…

Universal variance conjecture. For $\delta\in(0,1)$ and $H=X^\delta$,

Let $\Lambda(n)$ denote the von Mangoldt function. For $\delta\in(0,1)$, set $H=X^\delta$. Goldston–Montgomery conjecture. As $X\to\infty$, … This predicts the variance of sums of…

For $z>0$, let $d_z(n)$ be the generalized divisor function, define $D_z(x)=\sum_{n\leq x}d_z(n)$, and let $\overline D_z(x)$ be the contour approximation defined in the paper. Set…

Let $b(n)$ be the indicator of integers representable as sums of two squares. Let $B(x)=\sum_{n\leq x}b(n)$, let $I(x;H)$ be the paper's smooth approximation to $B(x+H)-B(x)$, and…

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 $\mu(n)$ denote the Möbius function and let $\omega(n)$ denote the number of distinct prime factors of $n$. For $X\to\infty$, let $H=X^\delta$ with $\delta\in(0,1)$. Short-inte…

Let $A$ be the polynomial ring over a finite field of size $q$. For a monic polynomial $f\in A$ of degree $d$ and a positive integer $m<\deg(f)$, let … For a partition $\lambda$ of…

Let $p_k$ denote the $k$th prime, let $s_k:=\{p_k^2,\dots,p_{k+1}^2-1\}$, and let $\pi_k$ be the number of primes in $s_k$. For $x$ and an interval length $y=y(x)$, write…