7 problems

Let $x\geq 1$, let $H$ be a real number, and let $\varepsilon\in(0,1/100)$. The function $\mu^2(n)$ is the indicator of squarefree integers. Short-interval conjecture. Uniformly fo…

For $x\geq 1$ and $h\geq 1$, define … where $\mu(n)$ is the Möbius function. Squarefree numbers in short intervals conjecture. For every $\varepsilon>0$, there exists…

Let $p\ge 3$ be prime and set $N_p=2^p-1$. The squarefreeness conjecture. There exists $p_0$ such that, if $p>p_0$, then $N_p$ is squarefree. Squarefreeness would rule out repeated…

Squarefree-gap conjecture. For any $\epsilon>0$ and $x$ sufficiently large in terms of $\epsilon$, the interval $(x,x+x^\epsilon]$ always contains a squarefree number.

Let $X\geq 1$ and let $x$ be drawn uniformly at random from $[X,2X]$. For a positive scale $H$, define the fluctuation process … Here $\mu^2(n)$ is the squarefree indicator. Fracti…

Let $x\geq 1$, let $h$ be an integer with $1\leq h\leq x^{1-\varepsilon}$, and let $\varepsilon>0$. Let $C(h)$ be a constant depending only on $h$. Squarefree correlation conjectur…

Let $r_\nu(N)$ denote the number of representations of $N$ as the sum of $\nu$ squarefree numbers. Suppose that … where $\mathfrak{S}_\nu(N)$ is the singular series and…