A new formula for the \(n\)-th prime number (Q2762944)

Let \(\pi(X)\) be the function defined as the number of prime numbers less than a real number \(X > 0\), and let \(p _n\) be the \(n\)-th prime number in the sequence of natural numbers. It is known that \(p _n \leq [(n ^2 + 3n + 4) /4]\) for each \(n \in {\mathbb N}\) [\textit{D. Mitrinović} and \textit{M. Popadić}, Inequalities in Number Theory. Niš, Univ. of Niš (1978; Zbl 0395.10001)].NEWLINENEWLINENEWLINEThe paper under review provides three exact formulae for \(\pi (n)\), and such a formula for \(p _n\). In particular, it has been proved that \(\pi (n) = \sum _{k=2} ^n \overline{\text{ sg}}(k - 1 - \varphi (k))\), and \(p _n = \sum _{i=0} ^{C(n)}{ \text{ sg}}(n - \pi (i))\), where \(C _n = [(n ^2 + 3n + 4)/4]\), \(\varphi \) is Euler's function, \(\text{ sg}\) is the characteristic function of the set of real positive numbers, \(\overline{\text{ sg}}(0) = 1\) and \(\overline{\text{ sg}}(x) = 0: x \neq 0\). The proofs are obtained as consequences of the definitions of the considered functions and some of their well-known properties, and the inequality \(p _n \leq C(n)\).