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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 \(m\geq 1\).

11A41 Primes
11N05 Distribution of primes
Full Text: DOI
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