On the estimates of the upper and lower bounds of Ramanujan primes

Yang, Shichun; Togbé, Alain

doi:10.1007/s11139-015-9706-8

Cited by 2 publications

(3 citation statements)

References 3 publications

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“…For any fixed positive integer m, we have, by (1.4), π(R mn ) ∼ 2mn ∼ mπ(R n ) as n → ∞. A first result in the direction of Conjecture 1.5 is due to Yang and Togbé [17,Theorem 1.3]. They used Proposition 1.1 to find the following result, which proves Conjecture 1.5 when n satisfies n > 10 300 .…”

Section: Introductionmentioning

confidence: 92%

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On the number of primes up to the $n$th Ramanujan prime

Axler¹

2017

Preprint

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The nth Ramanujan prime is the smallest positive integer Rn such that for all x ≥ Rn the interval (x/2, x] contains at least n primes. In this paper we undertake a study of the sequence (π(Rn)) n∈N , which tells us where the nth Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the nth Ramanujan prime, which imply an asymptotic formula for π(Rn) conjectured by Yang and Togbé. In the second part of this paper, we use these explicit estimates to derive a result concerning an inequality involving π(Rn) conjectured by of Sondow, Nicholson and Noe.

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Section: Introductionmentioning

confidence: 92%

“…To prove Theorem 1.2, we use the method investigated by Yang and Togbé [17] for the proof of the upper bound for π(R n ) given in Proposition 1.1. First, we note following result, which was obtained by Srinivasan [12,Lemma 2.1].…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

On the number of primes up to the $n$th Ramanujan prime

Axler¹

2017

Preprint

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“…The above result follows from a special case of our main theorem given below. Yang and Togbe [11], also used the method in [9], to give tight upper and lower bounds for R n for large n (greater than 10 300 ). For some interesting generalizations of Ramanujan primes the reader may refer to [2], [5] and [6].…”

Section: Introductionmentioning

confidence: 99%

New upper bounds for Ramanujan primes

Srinivasan

Arés-Gastesi

2018

Glas. Mat. Ser. III

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For n ≥ 1, the n th Ramanujan prime is defined as the smallest positive integer R n such that for all x ≥ R n , the interval ( x 2 , x] has at least n primes. We show that for every ǫ > 0, there is a positive integer N such that if α = 2n 1 + log 2 + ǫ log n + j(n), then R n < p [α] for all n > N , where p i is the i th prime and j(n) > 0 is any function that satisfies j(n) → ∞ and nj ′ (n) → 0.

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On the estimates of the upper and lower bounds of Ramanujan primes

Cited by 2 publications

References 3 publications

On the number of primes up to the $n$th Ramanujan prime

On the number of primes up to the $n$th Ramanujan prime

New upper bounds for Ramanujan primes

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