Is there a closed formula for the number of integer divisors?

Definition. Let $\Bbb R$ be the set of all real numbers. A closed formula on the set $\Bbb R$ (or its subset) is a finite (the number of elements does not depend on the value of the argument) combination and/or superposition of arithmetic operations and elementary functions — power, exponential, logarithmic, trigonometric, taking the integer/fractional part, etc.

Statement. Let $\Bbb N$ be the set of all natural numbers. Let $d(n)$ be a function of the number of all distinct natural divisors of $n$ defined on $\Bbb N$. There is no closed formula $F(x)$ on the set of all positive real numbers $\Bbb R_+ = \{x\in \Bbb R \mid x>0\}$ such that the restriction of $F(x)$ to $\Bbb N$ coincides with $d(n)$.

If you know that a given statement is strictly true or false, please provide a link to the proof.

The answer is no, so there exists a closed formula satisfying the requirements. Indeed, Prunescu and Sauras-Altuzarra showed that there is a closed formula for $d(n)$ that is built up from the binary operations $$x+y,\qquad \max(x-y,0)=\frac{\sqrt{(x-y)^2}+x-y}{2},\qquad \lfloor x/y\rfloor,\qquad x^y.$$