2 problems

Let $x\geq 2$ be large, let $r>0$ be real, and set $y=\log^r x$. With $\pi(t)=\#\{p\leq t:p\text{ is prime}\}$, the oscillation conjecture. Both inequalities … and … occur infinite…

Let $n>1$, let $d>1$ be a factor of … and let $q$ be the least prime factor of $d$. Prime-interval conjecture. There are primes in each of the intervals $(d-n,d+n]$ and $(d-q,d+q)$…