Estimates for the sequence of primes (Q1263607)

Let \(p_ n\) be the nth prime and let \(\pi\) (x) be the number of primes p such that \(p\leq x\). We prove by elementary methods that \[ 0.91 n\cdot \log n<p_ n<1.7 n\cdot \log n,\quad for\quad n\geq 3, \] \[ 0.788 x<\pi (x)\cdot \log x<1.5 x,\quad for\quad x\geq 5, \] and \[ 0.6 x/\log (2x)<\pi (2x)-\pi (x),\quad for\quad x\geq 6. \] These estimates are considerably sharper than those previously obtained by related methods. The basic idea is to work with certain multinomial coefficients instead of the binomial coefficient \(\binom{2n}{n}\) as it is usually done. This also enables us to prove that \[ 2.1^ n<\prod_{p\geq n}p<3^ n,\quad for\quad n\geq 41, \] which yields a one-line proof of Bertrand's postulate.