BOUNDS ON THE NUMBER OF PRIMES IN RAMANUJAN INTERVAL

The Ramanujan primes are the least positive integers Rn having the property that if m ≥ Rn, then πm − π(m/2) ≥ n. This document develops several bounds related to the Ramanujan primes, sharpening the currently known results. The theory presented is by no means exhaustive, however it provides insights for future research work. Alternatively, we may say that it is a road map which may be followed to make further discoveries. 2010 Mathematics Subject Classification. 0102, 1102, 1103, 11A25, 11A41, 11N05.

Note that R n is indeed a prime, by minimality condition the function π m −π (m/2) and π m must increase at m = R n . Since they can increase by at most 1, the equality π Rn − π (Rn/2) = n holds (Sondow [16]).

Bounds on s and p s on the Ramanujan Interval.
From the paper under the title "An improved upper bound for Ramanujan primes" by Srinivasan et al [19], we have the upper bound on s ∈ N: Let R n = p s be the n-th Ramanujan prime, where p s is the s-th prime. Then, for all n ∈ N | n ≥ 19, R 19 = 227: (1.1) s < 2n  and s. The asymptote is clearly visible between n = 18 and n = 19, the global minimum occurs at n = 50 where it attains approx. 219.095. The figure is drawn at every n ∈ N in the range. In this document we present improved bounds pertaining to the case R n = p s , where n, s ∈ N are ordinal numbers indicating the n th Ramanujan prime, and s th Figure 3. The drawing shows the estimation error made by the Supremum (Red) and Infimum (Blue) of the prime counting function π Rn . The figure is drawn at every R n ∈ N in the range.

Tailored Log Integral Bounds on π (Rn)
prime. Thus we have that in the case R n = p s , necessarily s = π (Rn) . In such a case we can improve the estimate of s very significantly by implementing the Tailored integral theorems by Feliksiak [3], which provide sharper estimates. The Tailored logarithmic integral definitions follow: where p is the prime p ∈ N | p ≤ R n and γ is the Euler-Mascheroni constant γ ≈ 0.57721566490153286060651209.

Theorem 2.3 (Supremum and Infimum Bound on s).
Let R n = p s be the n th Ramanujan prime, where p s is the s th prime. Then, for all n ∈ N | n ≥ 47: (2.1) Proof.
For the proof, please refer to Feliksiak [3].
The bounds given by Theorem 2.3, are computationally quite demanding for larger R n ∈ N. Consequently more efficient bounds are given by Theorem 2.6: where the prime R ∈ N | R n ≥ 47 Proof.
For the proof, please refer to Feliksiak [3].
3. Bounds on π (Rn) in terms of the ordinal number n There may be the need to express π (Rn) = s in terms of the ordinal number n. Therefore, elementary upper and lower bounds on s in terms of the ordinal number n are given by Theorem 3.1 (please also refer to Figures 6 and 7).  The bounds on π Rn = s, s.t. s ∈ N subject to p s = R n , in terms of the ordinal number n for all n ∈ N | n ≥ 2 is given by: Where p s is the s th prime number s.t. ∀n ∈ N | p s = R n and R n is given by the Definition 1.2 and the exponents are: α = 9947 10 000 and β = 10 008 10 000 The proof is limited to the proof of the Upper Bound only. The proof of the Lower Bound is left at the discretion of the reader.

Proof.
Suppose that the Upper Bound of Theorem 3.1 is false for n ∈ N | n ≥ 20 000. Then necessarily, Thus clearly, The Cauchy Root test of the absolute value of the difference 3.3 for n ∈ N | n ≥ 20 000 is: At n = 20 000 the root test 3.4 attains approximately 1.00023 and tends to 1 strictly from above. By the definition of the Cauchy Root test therefore, the sequence is diverging. In accordance with the hypothesis therefore, the difference 3.3 must diverge to −∞ as n increases unboundedly. However, at n = 20 000, the difference 3.3 attains approximately 99.1734 and diverges at an increasing rate as n increases unboundedly. Consequently, we have a contradiction to the initial Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 September 2021 doi:10.20944/preprints202109.0097.v1 hypothesis. Thus the difference 3.2 for n ∈ N | n ≥ 20 000 exceeds at every step √ n.
Necessarily, it must be true that the UB of Theorem 3.1 is true for n ∈ N | n ≥ 20 000. It is a straightforward exercise to verify that UB of Theorem 3.1 holds within the range 2 ≤ n ≤ 20 000 as well, please refer to Figure 8.
This implies that the Upper Bound of Theorem 3.1 holds for all n ∈ N | n ≥ 2, consequently this concludes the proof.   The upper bound on the prime number p s , where p s denotes the sequential s th prime number, for all n, s ∈ N | p s = R n ≥ 11 is given by: The proof is limited to the proof of the Lower Bound only. The analogous proof of the Upper Bound is left at the discretion of the reader.

Proof.
We begin with: Suppose that Theorem 4.1 is false for some p s ∈ N | p s ≥ 151, in accordance with the hypothesis this implies that: However, at p s = 151 the difference 4.3 attains approximately 3.7885 × 10 65 and further increases exponentially. Therefore, we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers p (s−1) , p (s) ∈ N| p (s) ≥ 151: The sequence a (s) given by the Definition 4.2, has the least value at the Ramanujan twin primes, since the difference p (s) − p (s−1) = 2. Consequently, it is therefore both necessary and sufficient, to consider the sequence a (s) at the Ramanujan twin primes only, with p (s) = 6i + 7 | i ∈ N, i ≥ 24.
At the Ramanujan twin primes: (2) Thus, at the Ramanujan twin primes the sequence a (s) equals: The bracketed expression on the RHS, at the Ramanujan twin primes approaches the limit: Therefore, at the Ramanujan twin primes, the sequence a (s) must clearly approach the limit: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 September 2021 doi:10.20944/preprints202109.0097.v1 By the definition of the d'Alemberts Ratio Test therefore, the difference exp ( p (s) ) − exp LB s , diverges as p (s) increases unboundedly. Thus, it logically follows that at Ramanujan twin primes: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the Ramanujan twin primes the sequence exp ( p (s) ) − exp LB s approaches: Rearranging the above, we obtain that at the Ramanujan twin primes exp LB s approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (s) attains, therefore this result holds for all p (s) ∈ N| p (s) ≥ 151. By taking the logarithms of both sides, we obtain:

Alternative Bound on R n on the Ramanujan Interval
This section develops a new set of bounds on R n w.r.t. n ∈ N in the Ramanujan Interval. These bounds constitute a significant improvement upon previously known bounds. The initial estimate presented by J. Sondow [16] had been further researched and optimized, producing bounds that follow R n quite closely. The upper and lower bounds on the Ramanujan prime number R n , w.r.t. the ordinal number n, which denotes both the sequential n − th Ramanujan prime number R n and the number of primes contained within the Ramanujan Interval { n 2 , n } , are given by: With α given by α = 599 625 and β is given by β = 601 625 . The proof is limited to the proof of the Lower Bound only. The proof of the Upper Bound is left at the discretion of the reader.

Proof.
We begin with: Suppose that Theorem 5.1 is false for some R n ∈ N | R n ≥ 151. In accordance with the hypothesis this implies that: However, at R n = 151 the difference 5.4 attains approximately 1.70404 × 10 993 and further increases exponentially. Therefore, we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers R (n−1) , R n ∈ N| R n ≥ 151: At the Ramanujan twin primes: (2) Thus, at the Ramanujan twin primes the sequence a (n) equals: The bracketed expression on the RHS, at the Ramanujan twin primes approaches the limit: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 September 2021 doi:10.20944/preprints202109.0097.v1 Therefore, at the Ramanujan twin primes, the sequence a (n) must clearly approach the limit: (2) By the definition of the d'Alemberts Ratio Test therefore, the difference exp ( R (n) ) − exp LB n , diverges as R (n) increases unboundedly. Thus, it logically follows that at Ramanujan twin primes: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the Ramanujan twin primes the sequence exp ( R (n) ) − exp LB n approaches: Rearranging the above, we obtain that at the Ramanujan twin primes exp LB n approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (n) attains, therefore this result holds for all R (n) ∈ N| R (n) ≥ 151. By taking the logarithms of both sides, we obtain: Thus, Theorem 5.1 holds for all R (n) ∈ N| R (n) ≥ 151. Direct computation verifies that Theorem 5.1 holds for all R (n) ∈ N | 11 ≤ R (n) ≤ 151. Please refer to Figures  11 and 12. Therefore, Theorem 5.1 holds as stated: (5.14) LB n < R (n) ∀n ∈ N | n ≥ 2 Concluding the proof of Theorem 5.1.

Remark 5.2.
This remark provides a few hints for the proof of the Upper Bound on R (n) : Equation 5.2. It can be used to further sharpen the Upper Bound, and help to facilitate an enhancement of the Lower bound in a similar fashion. Observe, that the corresponding ratio: at R n ≥ 11 it exceeds: Please also refer to Figures 13 and 14. Hence, the limit diverges: Consequently, by the definition of the d'Alemberts Ratio Test, exp (UB n )−exp ( R (n) ) diverges as R n increases unboundedly. This in turn implies that, The upper bound on R n as given by Laishram's Theorem (as quoted by Sondow et all [17]) in terms of the prime p (3n) is given by:

Theorem 5.4 (Sondow et al).
The maximum value of R n /p (3n) is:  In this paper we introduce an enhanced version of the upper and the lower bounds in terms of p (2n) , p (3n) on the Ramanujan Interval: Theorem 5.5 (Bounds on R n in terms of p (2n) , p (3n) ).
The bounds on R n for all n ∈ N | n ≥ 1, in terms of p (2n) and p (3n) are given by: With the exponent α given by 9921 10 000 .
The proof is limited to the proof of the Lower Bound only. The proof of the Upper Bound is left at the discretion of the reader.

Proof.
We begin with: Suppose that Theorem 5.5 is false for some R n ∈ N | R n ≥ 151. In accordance with the hypothesis this implies that: However, at R n = 151 the difference 5.20 attains approximately 3.7885 × 10 65 and further increases exponentially. Therefore, we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers R (n−1) , R n ∈ N| R n ≥ 151: Definition 5.6. a (n) = exp (Rn)−exp (LBn) exp (R(n−1))−exp (LB(n−1)) Remark 5.3. The sequence a (n) given by the Definition 5.6, has the least value at the Ramanujan twin primes, since the difference R (n) − R (n−1) = 2. Consequently, it is therefore both necessary and sufficient, to consider the sequence a (n) at the Ramanujan twin primes only, with R (n) = 6i + 7 | i ∈ N, i ≥ 24.
At the Ramanujan twin primes: (2) Thus, at the Ramanujan twin primes the sequence a (n) equals: The bracketed expression on the RHS, at the Ramanujan twin primes approaches the limit: Therefore, at the Ramanujan twin primes, the sequence a (n) must clearly approach the limit: (2) By the definition of the d'Alemberts Ratio Test therefore, the difference exp ( R (n) ) − exp LB n , diverges as R (n) increases unboundedly. Thus, it logically follows that at Ramanujan twin primes: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the Ramanujan twin primes the sequence exp ( R (n) ) − exp LB n approaches: Rearranging the above, we obtain that at the Ramanujan twin primes exp LB n approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (n) attains, therefore this result holds for all R (n) ∈ N| R (n) ≥ 151. By taking the logarithms of both sides, we obtain: (5.29) LB n < R (n) ∀R (n) ∈ N| R (n) ≥ 151 Thus, Theorem 5.5 holds for all R (n) ∈ N| R (n) ≥ 151. Direct computation verifies that Theorem 5.5 holds for all R (n) ∈ N | 2 ≤ R (n) ≤ 151. Please refer to Figures  15 and 16. Therefore, Theorem 5.5 holds as stated: (5.30) LB n < R (n) ∀n ∈ N | n ≥ 1 Concluding the proof of Theorem 5.5.