The Riemann Hypothesis (see also this paper) reflects the fact that all currently known approximations of the number $\pi(n)$ of primes $\leq n$ are heuristic

All known approximations of the number $\pi(n)$ of primes $\leq n$ are currently derived from real-valued functions that are asymptotic to $\pi(x)$, such as $\frac{x}{log_{e}x}$, $Li(x)$ and Riemann’s function $R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{(n)}li(x^{1/n})$, where the degree of approximation for finite values of $n$ is determined only heuristically by conjecturing upon an error term in the asymptotic relation that can be seen’, but not yet proven, to yield good’ approximations of $\pi(n)$ (as described, for instance, by Barry Mazur and William Stein on p.36 of their absorbingly written, but without sacrificing rigour, entry-level book Prime Numbers and the Riemann Hypothesis).

The above graph compares the actual number $\pi(x)$ (red) of primes $\leq x$ with the distribution of primes as estimated variously by the functions $Li(x)$ (blue), $R(x)$ (black), and $\frac{x}{log_{e}x}$ (green), where $R(x)$ is Riemann’s function $\sum_{n=1}^{\infty}\frac{\mu(n)}{(n)}li(x^{1/n})$.

Two non-heuristic approximations of the number $\pi(n)$ of primes $\leq n$

The following introduces, and compares, two non-heuristic prime counting functions (investigated more fully here) that, prima facie, should apparently yield square-root accurate approximations of $\pi(n)$:

(i) $\pi_{_{H}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}}) = n.\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}})$ (green); and

(ii) $\pi_{_{L}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{j}}(1-\frac{1}{p_{_{i}}})$ (red);

Based on manual and spreadsheet calculations, Figs.1 and 2 compare the values of the two non-heuristically estimated prime counting functions (both of which have binomial standard deviations) with the actual values of $\pi(n)$ (blue) for $4 \leq n \leq 1500$ and $4 \leq n \leq 3000$.

Fig.1: The above graph compares the non-heuristically estimated values of $\pi_{_{H}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}}) = n.\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}})$ (green) and $\pi_{_{L}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{j}}(1-\frac{1}{p_{_{i}}})$ (red) vs the actual values of $\pi(n)$ (blue) for $4 \leq n \leq 1500$.

Fig.2: The above graph compares the non-heuristically estimated values of $\pi_{_{H}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}}) = n.\prod_{i=1}^{\sqrt{n}}(1-\frac{1}{p_{_{i}}})$ (green) and $\pi_{_{L}}(n) = \sum_{j=1}^{n}\prod_{i=1}^{\sqrt{j}}(1-\frac{1}{p_{_{i}}})$ (red) vs the actual values of $\pi(n)$ (blue) for $4 \leq n \leq 3000$. Note that the gradient of both $y = \pi_{_{H}}(x)$ and of $y = \pi_{_{L}}(x)$ in the interval $(p_{_{n}}^{2},\ p_{_{n+1}}^{2})$ is $\prod_{i = 1}^{n}(1 - \frac{1}{p_{_{i}}}) \rightarrow 0$.

Three intriguing queries

Since $\pi(n) \sim \frac{n}{log_{e}(n)}$ (by the Prime Number Theorem), whilst $\pi_{_{H}}(n) \sim 2e^{-\gamma}\frac{n}{log_{_{e}}n}$ (by Mertens’ Theorem, where $\gamma$ is the Euler-Mascheroni constant and $2.e^{-\gamma} \approx 1.12292 \ldots$), and it can be shown (see Corollary 3.2 here)  that $\pi_{_{L}}(n) \sim ae^{-\lambda}\frac{n}{log_{_{e}}n}$ for some $a \geq 2$, this raises the interesting queries (see also $\S$2 of this initial investigation):

Fig.3. The overlapping rectangles $A, B, C, D, \ldots$ represent $\pi_{_{H}}(p_{_{j+1}}^{2}) = p_{_{j+1}}^{2}.\prod_{i = 1}^{j}(1 - \frac{1}{p_{_{i}}})$ for $j \geq 1$. Figures within each rectangle are the primes and estimated primes corresponding to the functions $\pi(n)$ and $\pi_{_{H}}(n)$, respectively, within the interval $(1,\ p_{_{j+1}}^{2})$ for $j \geq 2$.

Fig.4: Graph of $y = \prod_{i = 1}^{\pi(\sqrt{x})}(1 - \frac{1}{p_{_{i}}})$. The rectangles represent $(p_{_{j+1}}^{2} - p_{_{j}}^{2})\prod_{i = 1}^{j}(1 - \frac{1}{p_{_{i}}})$ for $j \geq 1$. Figures within each rectangle are the primes corresponding to the functions $\pi(n)$ and $\pi_{_{L}}(n)$ within the interval $(p_{_{j}}^{2},\ p_{_{j+1}}^{2})$ for $j \geq 2$. The area under the curve is $\pi_{_{L}}(x) = (x - p_{_{n}}^{2})\prod_{i = 1}^{n}(1 - \frac{1}{p_{_{i}}}) + \sum_{j = 1}^{n-1}(p_{_{j+1}}^{2} - p_{_{j}}^{2})\prod_{i = 1}^{j}(1 - \frac{1}{p_{_{i}}}) + 2$.

(a) Which is the least $n$, if any, such that $\pi_{_{H}}(n) > \pi(n)$?

(b) Which is the largest $n$, if any, such that $\pi(n) > \pi_{_{H}}(n)$?

(c) Is $\pi(n) / \frac{n}{log_{_{e}}n}$ an algorithmically verifiable, but not algorithmically computable (see Definitions 1 and 2 here), oscillating function which is discontinuous in the limit $n \rightarrow \infty$ (in the sense of Zeno’s paradox in 2-dimensions considered in Case 4 here)?