6 problems

Positive prime-power jump conjecture. Positive jumps of $\eta_N(x)$ occur at every prime power $x+1=p^l$, with $p\in\mathcal P$ and $l>1$. Moreover,

Negative-jump conjecture. Negative jumps of $\eta_N(x)$ occur at every point $x+1\in\mathcal P$, where $\mathcal P$ is the set of primes, and

Normalized Chebyshev-prime criterion. A Chebyshev prime of index $l$ is a prime $p_m$ satisfying the displayed inequality. This criterion is the formulation used to relate generali…

Prime-power midpoint conjecture. The jump satisfies

Midpoint jump conjecture. The jump at the prime $p_n$ satisfies

Universal jump conjecture. For every integer $n\geq 1$, one has $J_n>1$. The inequality is verified in the paper for $p_{n+1}<10^{11}$; its validity for all $n$ is proposed as a na…