4 problems

Let $S=(q_1,\dots,q_t)$ be a finite sequence of primes and define $\nu_S(n)=(\nu_{q_1}(n),\dots,\nu_{q_t}(n))$. Say that $n$ is $S$-squarefree if $q^2\nmid n$ for every prime…

For $X,Y\to\infty$ with $X\geq Y$, define … and let … where $(\Delta/p)$ is the Legendre symbol and $\Psi(X,Y)$ counts the $Y$-smooth integers at most $X$. Weak Granville conjectur…

For $X,Y\geq1$, let $P(n)$ be the greatest prime factor of $n$, and define … Let $\Psi(X,Y)$ denote the count of positive integers at most $X$ whose prime factors are all at most…

Chen–Chen conjecture. For any integer $k\ge 1$ and any $c\in [\frac{1}{k+1},\frac{1}{k})$, we have