I have been working with the Collatz Conjecture for fun and came about some interesting results. I know better than to claim this is a proof, but I would love if the community could share insight on my approach. We all know the problem, but I will re-write it here: $$ f(x) = \left\{ \begin{array}...

Theorem Let $f : \mathbb{N} \to \mathbb{C}$ be an arithmetic function and let $M(f, x) = \sum_{n \leq x} f(n)$ be the summatory function of $f$. If $M(f, x) = Ax^{\alpha} + O(x^{\theta})$, where $\alpha > \theta \geq 0$, then the Dirichlet series $F(s) = \sum_{n = 1}^{\infty} f(n)n^{-s}$ has a me...

I know this gets asked a lot, but I'd really appreciate comments about my proposed proof for the Collatz conjecture. At least to me it feels that demonstrating when an odd starting number goes to 1 occurs when p  ≥ ⌊log2[3q(x0+1)]⌋ as it does in my formulation of the problem seems like a novel f...