Let $a,b\in\mathbb{N}$ such that $(a,b)=1$. If $2\nmid a$, then one can show that $$\rho_{a/b}(n)=\sum_{k=1}^n (-1)^{\lfloor ak/b\rfloor}$$ is periodic. Is there any explicit way to give the maximum of $|\rho_{a/b}(n)|$ in terms of $a$ and $b$? As a first attempt we can show that $$\rho_{a/b}(n+b...

Let $a,b\in\mathbb{N}$ such that $(a,b)=1$. If $2\nmid a$, then one can show that $$\rho_{a/b}(n)=\sum_{k=1}^n (-1)^{\lfloor ak/b\rfloor}$$ is periodic. Is there any explicit way to give the maximum of $|\rho_{a/b}(n)|$ in terms of $a$ and $b$? As a first attempt we can show that $$\rho_{a/b}(n+b...

I ran across this post: Winter Bash 2018 Hat list On MSE (for all the secret hats), and I realized that the answer doesn't contain the explanation of Clean-Up duty hat. I am just wondering how you get it? On the chat of Winter Bash, they gave me an idea to re-tag some old questions and some ne...