7 problems

Let $T_c(x)$ denote the number of primes $p\le x$ such that the largest prime factor $P^+(p-1)$ satisfies $P^+(p-1)\ge p^c$, where $0<c<1$: … and let $\pi(x)$ be the prime-counting…

Shifted-prime moment conjecture. There exists a constant $c_k(a,b)>0$ such that

Pomerance's conjecture. For every $\theta\in(0,1)$,

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…

Let $\mathcal P(x,y)$ be the set of primes $p\leq x$ whose largest prime factor is at most $y$, and let $\Pi(x,y)=\lvert\mathcal P(x,y)\rvert$. Let $\pi(z)$ denote the number of pr…