The \(n\)-th prime asymptotically (Q2448537)

The authors give an asymptotic expansion for the n-th prime \(p_n\) along with explicit bounds for the error. The authors' main results are in terms of the function \(\operatorname{ali}:\mathbb{R}\to (1,\infty)\) defined by \(\operatorname{li}(\operatorname{ali}(x))=x\), i.e. \(\operatorname{ali}\) is the inverse of the well-known function \(\operatorname{li}(x) = \operatorname{PV}\int_{0}^{x}\frac{dt}{\log t}\), where PV means the principal value of this function. Among other results, the authors prove that the Riemann hypothesis is equivalent to the following inequality: \[ |p_n -\operatorname{ali}(n) | < \frac{1}{\pi}\sqrt{n} \log^{5/2} n,\qquad \forall n \geq 11. \] Besides, the authors present a variety of inequalities involving \(\operatorname{ali}(x)\) and \(p_n\).