New upper bounds for Ramanujan primes

Abstract

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.

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

“…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]. Although it is a direct consequence of the definition of a Ramanujan prime, it plays an important role in the proof of the upper bound for π(R n ) in Proposition 1.1.…”

“…Applying Theorem 4 from the paper of Sondow, Nicholson and Noe [11], we get a refined upper bound for the number of primes less or equal to π(R n ), namely that the inequality π(R n ) ≤ π(41p 3n /47) holds for every positive integer n with equality at n = 5. Srinivasan [12,Theorem 1.1] proved that for every ε > 0 there exists a positive integer…”

“…for every positive integer n ≥ N , where j is any positive function satisfying j(n) → ∞ and nj ′ (n) → 0 as n → ∞. Setting ε = 0.5 and j(n) = log log n − log 2 − 0.5, they found [13,Corollary] that the inequality (1.7) holds for every positive integer n ≥ 44. In 2016, Yang and Togbé [17, Theorem 1.2] established the following current best upper and lower bound for π(R n ) when n satisfies n > 10 300 .…”