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).

PiAndOthers

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.

Primes_less_than_n_1500_A

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.

Primes_less_than_n_3000

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 \S2 of this initial investigation):

Fig_3

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

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)?

Bhupinder Singh Anand

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