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Explicit estimate on primes between consecutive cubes. (English) Zbl 1201.11111

In 1930, G. Hoheisel [S.-ber. Akad. Berlin 1930, 580–588 (1930; JFM 56.0172.02)] proved that there is at least one prime in the short interval \([x, x + x^{\theta}]\) with \(\theta = 1 - (1/33,000)\), for sufficiently large \(x\). In 1940, A. E. Ingham [Q. J. Math., Oxf. Ser. 11, 291–292 (1940; Zbl 0025.02704 and JFM 66.0341.02)] showed that there is at least one prime in \([x, x^{3/5 + \varepsilon}]\), where \(\varepsilon \to 0^{+}\) as \(x \to \infty\), for sufficiently large \(x\). This implies that there is at least one prime between two consecutive cubes if the numbers concerned are large enough. In the present article, the author gives an explicit form of Ingham’s theorem and then proves that there is at least one prime between consecutive cubes if the numbers concerned are larger than the cubes of \(e^{e^{15}}\). This is achieved by first estimating the number of zeros in the right half-plane \(\sigma > 1/2\) for the Riemann zeta-function.
The first main result of the paper can be summarized as follows: Let \(N(\sigma, T)\) be the number of zeros \(\rho = \beta + i\gamma\) of the zeta-function for which \(\beta \geq \sigma\) and \(0 < \gamma \leq T\). If \(5/8 \leq \sigma < 1\) and \(T \geq e^{e^{18}}\), then \[ N(\sigma, T) \leq (453472.54) T^{8 (1 - \sigma)/3} \log^5 T. \] Using this estimate, an explicit form of Landau’s approximate formula for \(\psi(x)\), and a zero-free region for the zeta-function established by Ford in 2000, the author proves the second main result: If \(x \geq e^{e^{15}}\) and \(h \geq 3 x^{2/3}\), then \[ \psi(x + h) - \psi(x) \geq (h / \log x) (1 - \varepsilon(x)), \] where \[ |\varepsilon(x)| = (3192.34) \exp\left(-\frac{1}{273.79} \left(\frac{\log x}{\log\log x}\right)^{1/3}\right). \] From this follows the third main result: If \(x \geq e^{e^{45}}\) and \(h \geq 3 x^{2/3}\), then \[ \pi(x+ h) - \pi(x) \geq h\left(1 - (3192.34) \exp\left(-\frac{1}{283.79} \left(\frac{\log x}{\log\log x}\right)^{1/3}\right)\right). \] As a corollary, the author points out that: If \(x \geq e^{e^{15}}\), then there is at least one prime between each pair of consecutive cubes \(x^3\) and \((x + 1)^3\).

MSC:

11Y35 Analytic computations
11N05 Distribution of primes
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Online Encyclopedia of Integer Sequences:

Number of primes between n^3 and (n+1)^3.

References:

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