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On the estimates of the upper and lower bounds of Ramanujan primes. (English) Zbl 1343.11008
Summary: For $$n\geq 1$$, the $$n$$th Ramanujan prime is defined as the least positive integer $$R_{n}$$ such that for all $$x\geq R_{n}$$, the interval $$(\frac{x}{2}, x]$$ has at least $$n$$ primes. Let $$p_{i}$$ be the $$i$$th prime and $$R_{n}=p_{s}$$. Sondow, Laishram, and other scholars gave a series of upper bounds of $$s$$. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow-Nicholson-Noe [J. Sondow et al., J. Integer Seq. 14, No. 6, Article 11.6.2, 11 p. (2011; Zbl 1229.11014)] and prove that if $$n>10^{300}$$, then $$\pi (R_{mn})\leq m\pi (R_{n})$$ for $$m\geq 1$$.

##### MSC:
 11A41 Primes 11N05 Distribution of primes
OEIS
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##### References:
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