Problems & Puzzles: Puzzles

Puzzle 336. sigma(n)<prime(n) Faride Firoozbakht asks: How we can prove "for n>4, sigma(n)<prime(n)"?

---

Dan Dima wrote (November 4, 2005)

This is a sketch of proof.
It is quite easy to prove this statement: "for n > 4, sigma(n) < prime(n)",
using an intermediate inequality like:

"for n > N, sigma(n) < C * n * ln n < prime(n)", where N is some fixed integer and C a positive real constant as C(N).

We have to check the inequality when 4 < n <= N in order to complete the proof.

A note for the 1st inequality: it should be even stronger if the Riemann Hypothesis holds - known as the Robin criterion: sigma(n) < C * n * ln (ln n)
but we need only: sigma(n) < C * n * ln n.

For the 2nd inequality: C * n * ln n < prime(n), use the behavior of the inverse function of prime(n), the prime counting function pi(n) bounded by:
n / ln n < pi(n) < 1.25506 * n / ln n, if n >= 17. pi(n) ~ n / ln n.

We want: pi(floor(n * ln n)) < n, which is true because:
pi(floor(n * ln n)) < (n * ln n) / (ln (n * ln n)) < n.
 

***

---