On a Conjecture on Ramanujan Primes

Abstract

Abstract. For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if) ≥ n where π(ν) is the number of primes not exceeding ν for any ν > 0 and ν ∈ R. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime π and θ functions due to Dusart.

Ramanujan and Labos primes, their generalizations and classifications of primes

“…In [15], Sondow obtained some estimates for R n and, in particular, proved that, for every n > 1, R n > p 2n . Laishram [6] proved that R n < p 3n (a short proof of this result follows from a general Theorem 6 of the present paper, see Remark 2). Further, Sondow proved that, for n → ∞, R n ∼ p 2n .…”

“…In case v = 2, k = 3, by Theorem 6, we find R n = R 2 (n) < p 3n , for n ≥ 22398. A simple computer verification for n < 22398 leads to the Laishram result[6].…”

New upper bounds for Ramanujan primes

“…Let R n = p s , where p i denotes the i th prime. Sondow [7] showed that p 2n < R n < p 4n for all n, and conjectured that R n < p 3n for all n. This conjecture was proved by Laishram [4], and the upper bound p 3n improved by various authors ( [1], [8]). Subsequently, Srinivasan [9] and Axler [1] improved these bounds by showing that for every ǫ > 0, there exists an integer N such that R n < p [2n(1+ǫ)] for all n > N. * Using the method in [9] (outlined below), a further improvement was presented by Srinivasan and Nicholson, who proved that s < 2n 1 + 3 log n + log(log n) − 4 for all n > 241.…”

Generalized Ramanujan Primes

“…In 1919, Ramanujan [Ra19] proved a result which implies that R n exists, and he gave the first five Ramanujan primes as R n = 2, 11, 17, 29, 41, for n = 1, 2, 3, 4, 5, respectively. The case R 1 = 2 is Bertrand's Postulate (proved by Chebyshev): for all x ≥ 2, there exists a prime p with 1 2 x < p ≤ x. Sondow proved that R n ∼ p 2n as n → ∞ (where p m is the mth prime), and he and Laishram [La10] proved the bounds p 2n < R n < p 3n , respectively, for n > 1.…”