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
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)?