An explicit result for primes between cubes
| Autor: | Adrian W. Dudek |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
General Mathematics
Mathematics::Number Theory 010102 general mathematics Prime number prime numbers 010103 numerical & computational mathematics 11Y35 Term (logic) Legendre's conjecture 01 natural sciences Prime (order theory) Riemann zeta function Combinatorics Riemann hypothesis symbols.namesake Number theory 11N05 symbols 0101 mathematics Mathematics Riemann zeta-function |
| Zdroj: | Funct. Approx. Comment. Math. 55, no. 2 (2016), 177-197 |
| Popis: | We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function 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 $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers. |
| Databáze: | OpenAIRE |
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