Some Conjectures on the Divisor Function

Abstract

: We propose the following conjecture on ( ) n σ the sum-of-divisors function: log( n σ− will increase strictly and converge to 1 when n runs from the colossally abundant numbers to infinity. This conjecture is a sufficient condition for the Riemann hypothesis by Robin’s theorem, and it is confirmed for n from 4 10 up to 103078 10 . Further, we present two additional conjectures that are related to Robin’s theorem.