New upper bounds for Ramanujan primes

Sa？etak For n≥ 1, the nth 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 ε>0, there is a positive integer N such that if α=2n(1+(log 2+ε)/(log n+j(n))), then Rn< p[α] for all n>N, where pi is the ith prime and j(n)>0 is any function that satisfies j(n)→ ∞ and nj'(n)→ 0