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We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes x 3 and (x + 1) 3 if log log x ≥ 15.2000 Mathematics Subject Classification. 11Y35, 11N05.

This paper gives an explicit zero-free region for the Riemann zeta-function derived from the Vinogradov-Korobov method. We prove that the Riemann zeta-function does not vanish in the region σ ≥ 1 − .00105 log −2/3 |t| (log log |t|) −1/3 and |t| ≥ 3.

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