Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of (∆, x 0 ) such that for all x ≥ x 0 there exists at least one prime in the interval (x(1 − ∆ −1 ), x].

We provide an explicit O(x/T ) error term for the Riemann-von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem, an inequality of Ramanujan, and a result due to Cramér on gaps between primes.

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This paper updates two explicit estimates for primes between consecutive powers. We find at least one prime between n 3 and (n + 1) 3 for all n ≥ exp(exp(32.9)), and at least one prime in (n 296 , (n + 1) 296 ) for all positive n. These results are in part obtained with a explicit version of Goldston's (1983) estimate for the error in the Riemann-von Mangoldt explicit formula.