With regards to my research (connecting character degrees' arithmetic structure with the corresponding group's structure), I find myself in the situation (when studying the symmetric group $S_n$) of wanting the interval $(\frac 56n,n)$ to contain at least two primes. In the "Better results" section of the Wikipedia article "Bertrand's postulate" https://en.wikipedia.org/wiki/Bertrand%27s_postulate, several results can be found which can ultimately be used to calculate explicitly a minimal value $n_0$ so that, for all $n\geq n_0$, the stated interval necessarily contains two primes. Specifically, each of the last five results listed there could be used for this purpose. However, the results themselves seem like massive overkill (on the order of using a nuclear bomb to remove a hornets nest) while also leaving upwards of 2 million or more of the smaller $n$ values in need of further attention.

With this as backdrop, I am looking for a better way. Are there less in-depth results which can be used to show the intervals $(\frac{5n}6,n)$ contain more than one prime? References would prove most helpful, but short-ish proofs, especially ones not invoking indepth analysis, would also be appreciated; what I don't want is for a paper ostensibly focusing on character degrees to be bogged down in number theoretic details any more than is absolutely necessary.

For what it's worth, the correct $n_0$ appears to be 32 $60$. A computer search returned this, and also showed a (not surprising) general trend that the number of primes in the interval is generally, though not strictly, increasing as $n$ increases.

History: I am a group theorist by nature, working in character theory, and have had very, very little interaction with the Riemann zeta function.

Aside: while on the subject, I have a follow on question (whose answer I expect to be a comment). When authors writing in this area use $\text{ln}^2(x)$, do they mean $(\text{ln}(x))^2$ or $\text{ln}(\text{ln}(x))$?