TITLE: Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
QUESTION [7 upvotes]: Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$
Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $1$-periodic function $p:[n,\infty)\to\mathbb{R}$ such that
$$f(x) = - \frac{1}{2} + \frac{\pi}{4}x + \frac{p(x)}{\sqrt{x}}$$ for every $x\geq n$.
Question: Can this be true?

Edit (26.12.2021) Below is the portion of the graph of $p$.

REPLY [4 votes]: This can not be true: $p(m+1)=p(m)$ for an integer $m$ yields that $\pi$ is Algebraic.