De Factorisatione Numerorum I : In Pursuit of the Erymanthian Boar

We introduce a novel glance at factoring. The technique broached here departs from any known (at least to the author) factoring method. In this paper, we show, given a product of two large primes $N$ (a RSA modulus), how to select a multiplicative function $\sigma_k$ (dependent on $N$) related to the sum of divisors function and produce a nontrivial small linear relation among $\exp(\log^\epsilon N)$ values of $\sigma_k(n)$ for $|n-N| = O(\exp(\log^\epsilon N))$, (subject to a plausible conjecture). The tools to achieve this don't go beyond classical analytic number theory, as known one hundred years ago.