Let $h(X) = X/2$, and $f(X) = (3X + 1)/2$. Then clearly every iteration $g^i(X), X \in \Bbb{Z}$ the Collatz mapping $$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \ X = 1\pmod 2 \end{cases}$$ can be modelled by compositions of $f, h$. For example, $g^7(3) = 1 = hhhff(3)$ where $g(...

We know that is sufficient to prove the Collatz conjecture for the odd number case or iow $x \in 2\Bbb{Z} + 1$. I read somewhere that it was sufficient to prove Collatz for the case $x \in 6\Bbb{Z} + 2$, but I can't find it and I would also like a more official reference. I'm not finding anythin...

Why $\sum_{\gamma>0} (\frac{sin(\gamma/N)}{\gamma/N})^{2} = \mathcal{O} (NlogN)$, $\gamma$ is the imaginary part of the zeros of $\zeta (x)$. In Monthomery's Multiplicative Number threory I.Classical theory, chapter 15 ,p479, in the proof of littlewood's result on the error term of the prime numb...