Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$

Question

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?

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

Answer 1

Use the Euler-MacLaurin formula, $$\sum_{k=1}^\infty F(k)=\int_0^\infty F(k)\,dk+\tfrac{1}{2}[F(\infty)-F(0)]+\int_0^\infty (k-\text{Int}\,[k]-\tfrac{1}{2})F'(k)\,dk.$$

In this case $F(k)=\sqrt{\max\{1 -k^2/x^2,0\}}$, hence$^\ast$ $$f(x)=\sum_{k=1}^\infty F(k)=\frac{\pi x}{4}-\frac{1}{2}+(3-4\delta x)\frac{\sqrt {2\delta x}}{6\sqrt{x}}-\frac{0.075}{\sqrt{x}}-\frac{1}{12x^2}\,\text{Int}\,[x]+{\cal O}(1/x^{3/2}),$$ where $\text{Int}\,[x]$ denotes the integer part of $x$ and $\delta x=x-\text{Int}\,[x]$. (The coefficient $0.075$ is computed numerically -- can it be computed analytically?)

So the equation in the OP, $$f(x) = - \frac{1}{2} + \frac{\pi}{4}x + \frac{p(x)}{\sqrt{x}},$$ with $p(x+1)=p(x)$ holds, but only up to corrections of order $1/x$.

The plot compares the exact $p(x)$ (blue) with expansion formula above (gold).

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_{$^\ast$ The integral of $k-\text{Int}\,[k]-\tfrac{1}{2}F'(k)$ over an interval $(n,n+1)$ contributes $-\tfrac{1}{12}x^{-2}$ for large $x$, and there are $\text{Int}\,[x]$ of these intervals, hence the term of order $1/x$ that breaks the periodicity of $p(x)$. The interval $(\text{Int}\,[x],x)$ contributes the term $\tfrac{1}{6}(3-4\delta x)\sqrt{2\delta x/x}$ for large $x$. The term $-0.075x^{-1/2}$ comes from the intervals $(n,n+1)$ with $n$ of order $x$. This is the contribution which I was not able to computer analytically.}

Answer 2

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