On generalized Ramanujan primes

Axler, Christian

doi:10.1007/s11139-015-9693-9

Cited by 7 publications

(3 citation statements)

References 11 publications

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“…In this section we give a proof of Theorem 1.7 by using Theorem 3.22 of [1]. For this, we need to introduce the following notations.…”

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 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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“…In this section we give a proof of Theorem 1.7 by using Theorem 3.22 of [1]. For this, we need to introduce the following notations.…”

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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“…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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“…In 2009, Mitra, Paul, and Sarkar [5] had generalized the Bertrand's postulate to conjecture that for any integers n, a and k, where a = ⌈1.1 ln(2.5k)⌉, there are at least k − 1 primes between n and kn when n ≥ a. In 2016, Christian Axler provided a proof [4] of this conjecture using an application of generalized Ramanujan primes. However, his proof works for sufficiently large positive integers.…”

Section: Introductionmentioning

confidence: 99%

Revisiting Generalized Bertand's Postulate and Prime Gaps

Das¹,

Paul²

2017

Preprint

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It is a well-known fact that for any natural number n, there always exists a prime in [n, 2n]. Our aim in this note is to generalize this result to [n, kn]. A lower as well as an upper bound on the number of primes in [n, kn] were conjectured by Mitra et al. [Arxiv 2009]. In 2016, Christian Axler provided a proof of the lower bound which is valid only when n is greater than a very large threshold. In this paper, after almost a decade, we for the first time provide a direct proof of the lower bound that holds for all n ≥ 2. Further, we show that the upper bound is a consequence of Firoozbakht's conjecture. Finally, we also prove a stronger version of the bounded gaps between primes.

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On generalized Ramanujan primes

Cited by 7 publications

References 11 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

Revisiting Generalized Bertand's Postulate and Prime Gaps

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