# Ramanujan primes in short intervals

I'm curious to know if it is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia Ramanujan prime or ) in short intervals than those that refers the Wikipedia Bertrand's postulate for prime numbers due to Pierre Dusart or Baker, Harman and Pintz:

There is at least a Ramanujan prime $$R$$ in the interval $$x or, say, in the interval $$x−x\cdot f(x)\leq R\leq x,\tag{2}$$ for a suitable function $$f(x)$$, and for all $$x\geq x_0$$ being $$x_0$$ a suitable constant (it is your choice the value of this constant).

Question. If you know it from the literature please refer, answering as a reference request, the literature and I try to search and read those statements from the literature. In case that isn't in the literature, can you provide a statement for Ramanujan primes in short intervals of the form $$(1)$$ or $$(2)$$? Many thanks.

I think that this is an interesting question, please feel free to add your feedback in comments.

## References:

 Jonathan Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly, Vol. 116, No. 7 (2009), pp. 630-635.

• The answer to all your Ramanjuan primes questions is the same: Ramanjuan primes have no useful meaning, no practical generating function, thus no theorem. What can be proven about primes is in the (analytic and algebraic) number-theory books. Mar 12, 2020 at 20:41
• I will not interact with your comments. Of course my respect and thanks for all your past and future contributions @reuns . An aside of your opinion, as you see yourself comment don't add any response to my specific Question. Best wishes and many thanks again. Mar 12, 2020 at 20:46
• All users I add a (unrelated motivation here, but maybe spiring the importance of Ramanujan primes) comment: that issues about Ramanujan primes are focused and arosed in analytic number theory, although it is my belief that literature is proliferating in recent past years about applications in other fields (for example I mean to search in a browser of Internet key words as Ramanujan primes, entropy, image analysis). It seems to me the same phenomenon than other well-known constellations of primes, let's say than these have applications in technology. I hope don't disturb with the comment Mar 18, 2020 at 8:14
• And I don't add all the references that I know about Ramanujan primes published in these years. Mar 30, 2020 at 14:07

According to Wikipedia, for $$n>1$$ we have $$p_{2n}, where $$p_n$$ is the $$n$$-th prime number. But $$p_{2n}>2n(\log (2n)+\log\log (2n)-1)$$ and $$p_{3n}<3n(\log (3n)+\log\log (3n))$$ for $$n\ge 3$$, see Wikipedia. Thus if $$x\le 2n(\log (2n)+\log\log (2n)-1)$$ then between $$x$$ and $$3n(\log (3n)+\log\log (3n))$$ there is a Ramanujan prime. This should provide a bound $$f(x)\approx\tfrac 12$$.