Normalized Chebyshev criterion for generalized Chebyshev primes

Let $\psi_0(x)$ be the normalized Chebyshev function, equal to $\psi(x)$ away from prime powers and with half the jump removed at a prime power. For a prime $p_m$ and an index $l\geq1$, define a Chebyshev prime of index $l$ by the inequality

$$\psi_0(p_m^l)>p_m^l.$$

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 generalized Chebyshev primes to oscillations of the Chebyshev function; its broader consequences remain conjectural in the paper.