On the sum of positive divisors functions

Abstract

AbstractProperties of divisor functions $$\sigma _k(n)$$ σ k ( n ) , defined as sums of k-th powers of all divisors of n, are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at $$x=0$$ x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions.