Schoenberg proved that for every $c\in [0,1]$ the density of$$\{n\in \mathbb{N}:\varphi (n)<cn\}$$exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f^{\prime }(x)$ exists and is positive?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let [ f(c)=\lim_{x\to\infty}\frac1x|\\{n\le x:\ \varphi(n)<cn\\}|\qquad(c\in[0,1]), ] so $f$ is the limiting distribution function of (\varphi(n)/n) [[nomath]](the difference between “$<$” and “$\le$” is irrelevant at continuity points, and in fact $f$ is known to be continuous)[[/nomath]].

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

4 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.