Abstract
We prove that there is a prime between n3 and (n + 1)3 for all n ≥ exp(exp(33:3)). This is done by first deriving the Riemann-von Mangoldt explicit formula for the Riemann zetafunction with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive mth powers for m ≥ 5×109. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.
Original language | English |
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Pages (from-to) | 177-197 |
Number of pages | 21 |
Journal | Functiones et Approximatio, Commentarii Mathematici |
Volume | 55 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2016 |