Short effective intervals containing primes (Q1869779)

Useful explicit estimates for the prime counting function were given by \textit{J. B. Rosser} and \textit{L. Schoenfeld} [Ill. J. Math. 6, 64--94 (1962; Zbl 0122.05001)]. In the study of prime gaps, it is more convenient to have explicit bounds for an interval which will contain a prime, and some such results based on calculations involving the zeros of the Riemann zeta-function are also given by \textit{L. Schoenfeld} [Math. Comput. 30, 337--360 (1976; Zbl 0326.10037)]. The authors prove that, under the Riemann hypothesis, the interval \((x-{8\over 5}\sqrt x\log x,x]\) always contains a prime when \(x>2\). Unconditional results are given in the form of a pair \((x_0,\Delta)\) with the property that the interval \((x-x/\Delta,x]\) contains a prime when \(x> x_0\). A table of values for such pairs is given; thus Theorem 3 states that \(x_0=10 727 905 041\) and \(\Delta=28 314 000\) form a valid pair. Here \(x_0\) is a prime with the corresponding prime gap 381 (largest before \(x_0\)) and \(x_0/\Delta=378{\cdot}8\ldots \). As a corollary, it is verified that odd numbers up to \(10^{22}\) are sums of at most three primes. The analysis involves the use of a smooth non-negative weight function associated with the interval concerned, which then enables the authors to employ an upper bound sieve and the Brun-Titchmarsh inequality to show that the detected primes are not clustering near the boundary in order to establish the results.