New upper bounds for ramanujan primes

Abstract

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For n ≥ 1, the n th Ramanujan prime is defined as the smallest positive integer Rn such that for all x ≥ Rn, the interval ( x 2 , x] has at least n primes. We show that for every > 0, there is a positive integer N such that if α = 2n

1 + log 2 + log n + j(n)

, then Rn < p[α] for all n > N, where pi is the i th prime and j(n) > 0 is any function that satisfies j(n) → ∞ and nj0 (n) → 0