**The Riemann Hypothesis (see also this paper) reflects the fact that all currently known approximations of the number of primes are heuristic**

All known approximations of the number of primes are currently derived from real-valued functions that are asymptotic to , such as , and Riemann’s function , where the degree of approximation for finite values of 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 (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 (red) of primes with the distribution of primes as estimated variously by the functions (blue), (black), and (green), where is Riemann’s function .

**Two non-heuristic approximations of the number of primes **

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 :

(i) (green); and

(ii) (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 (blue) for and .

Fig.1: The above graph compares the non-heuristically estimated values of (green) and (red) vs the actual values of (blue) for .

Fig.2: The above graph compares the non-heuristically estimated values of (green) and (red) vs the actual values of (blue) for . Note that the gradient of both and of in the interval is .

**Three intriguing queries**

Since (by the Prime Number Theorem), whilst (by Mertens’ Theorem, where is the Euler-Mascheroni constant and ), and it can be shown (see Corollary 3.2 *here*) that for some , this raises the interesting queries (see also 2 of *this initial investigation*):

Fig.3. The overlapping rectangles represent for . Figures within each rectangle are the primes and estimated primes corresponding to the functions and , respectively, within the interval for .

Fig.4: Graph of . The rectangles represent for . Figures within each rectangle are the primes corresponding to the functions and within the interval for . The area under the curve is .

(a) Which is the least , if any, such that ?

(b) Which is the largest , if any, such that ?

(c) Is an *algorithmically verifiable, but not algorithmically computable* (see Definitions 1 and 2 *here*), oscillating function which is *discontinuous in the limit* (in the sense of Zeno’s paradox in 2-dimensions considered in Case 4 *here*)?

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