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 $\text{ln}(\text{ln}(x))$?
This note, addressing @Peter Humphries questions, should be a comment, but is way too long to be recorded as such.
Why not just cite a paper that does the work for you?
I remind you, this question asked for, and in fact received, references. Furthermore, the Molsen paper given in @Gerhard's response exactly fits the bill; I am citing a paper, and in fact a paper which leaves me with far fewer sporadic cases with which to contend.
If it's for a research paper, why not kill a mosquito with a nuke if you can? I don't think the reader will be so bothered that you haven't used the simplest tools.
This question has multiple answers to it.
(1) There is an underlying philosophy, one I had drilled into me during grad school, that it is a very slippery slope to quote results without understanding them. Ideally, this means having digested all proofs involved in their entirety, including all relevant references, recursively working back to the basics, before I am willing to reference them in my work. When this does not happen, how much confidence do I myself have in what I am writing? If I do not know the result I am quoting, why should I believe the conclusion I'm asserting? I don't need to publish for my job, I just like to do it, so am under no pressure to publish. Without that pressure, in order to be willing to publish a paper, I need to be comfortable with all assertions in the paper, and therefore with the results I'm quoting.
Of course, this leads to a confrontation between the ideal and the practical, but it is exactly trying to straddle the horns of that particular dilemma that caused me to post this question in the first place. (Put another way, the question was really "Could somebody maybe, pretty please, give me a more readable reference than the ones I have thus far found, as I am way out of my league?")
Within this paradigm, using simpler tools has multiple advantages (a) It takes me, and the audience, less time to understand the material. (b) I, and the audience, are more comfortable with the validity of the argument. (c) A simpler result, proven both directly in one place and as a consequence of a deep theorem proven somewhere else, has a much greater chance of being correct, in spite of the fact that we are all human and therefore fallible.
And, I remind you that I already admitted not to having had much of any interaction with the Riemann zeta function. I wrote that to point out that I am very, very unfamiliar with this research area, making these points all the more poignant.
While I'm not afraid to reach out into the unknown, why should I force myself to do so when I have a readily-available alternative? So, let's get back to the question "...why not kill a mosquito with a nuke...?" Because while it is true that someone handed me a nuke, it is also true someone else handed me a flyswatter.
(2) The second reason is deeply tied to axiomatic systems on the whole. It has not been proven that the axioms of mathematics (e.g., ZFC+AC) are in fact self-consistent. What if a flaw is found and they in fact aren't? If these 13 axioms (say) are shown to be mutually inconsistent, how much of mathematics can be recreated using some particular subset of 7 axioms, ones which have not (yet) been proven inconsistent? Even should the axioms themselves be consistent, human fallibility still creeps in, so that the stated results themselves, though believed valid, either aren't valid or at least haven't been proven valid.
This leads me to another personal philosophy, a philosophy put succinctly by a quote attributed to Albert Einstein. "Everything should be made as simple as possible, but not simpler." We scientists love to focus on the second part, too often forgetting the first. The more we work towards making our results as simple as possible, the more easily we will recover from whatever quagmire may eventually come our way should things happen to fall apart. Which bring us to "...why not kill a mosquito with a nuke...?" Because the fallout can be very unpredictable, widespread, and hard to clean up.
