Known bounds on |π(x) − Li(x)| imply p_n lies between Li⁻¹(n(1 − a f(n ln n))) and Li⁻¹(n(1 + a f(n ln n))), but the stated n* is too small for large-a bounds.
The n-th prime asymptotically
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with $\li^{-1}(n)$, after having retained the first m terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n> s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion.
fields
math.NT 1years
2025 1verdicts
REJECT 1representative citing papers
citing papers explorer
-
The n-th prime exponentially
Known bounds on |π(x) − Li(x)| imply p_n lies between Li⁻¹(n(1 − a f(n ln n))) and Li⁻¹(n(1 + a f(n ln n))), but the stated n* is too small for large-a bounds.