**1 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 .

**1A 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, yield *non-heuristic* 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 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 .

**1B 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*), function which is *discontinuous in the limit* (in the sense of Zeno’s paradox in 2-dimensions considered in Case 4 *here*)?

**1C Three intriguing observations**

Fig.5: The above graph compares the actual values of in the range , with in the range , and in the range .

Fig.6: The above graph compares the actual values of in the range , with in the range , and in the range .

Fig.7: The above graph compares the actual values of in the range , with in the range , and in the range .

Fig.8: The above graph compares the actual values of in the range , with in the range , and in the range .

Fig.9: The above graph compares the actual values of in the range , with in the range , and in the range .

Query (a) is answered by Figs.8-9, which show that the least such that occurs somewhere around .

As regards Query (b) however, we conjecture that Fig.9 suggests that there is no larger such that .

Finally, we conjecture that—despite what is suggested by the Prime Number Theorem—Query (c) can be answered affirmatively if the three functions , , and are as well-behaved as Fig.9 suggests!

**2 Density of integers not divisible by primes **

The above observations reflect the circumstance that (formal proofs for the following are detailed in *Chapters 34 and 35 here*):

**Lemma 2.1**: The asymptotic density of the set of all integers that are not divisible by any of a given set of primes is:

.

It follows that:

**Lemma 2.2**: The expected number of integers in any interval that are not divisible by any of a given set of primes is:

.

**2A The function **

In particular, the expected number of integers that are not divisible by any of the first primes is:

**Corollary 2.3**:

It follows that:

**Corollary 2.4**: The expected number of primes is:

We conclude that is a *non-heuristic* approximation of the number of primes :

**Lemma 2.5**: .

**2B The function **

We note similarly that:

**Corollary 2.6**: The expected number of primes in the interval () is:

.

It further follows from Lemma 2.2 and Corollary 2.6 that:

**Corollary 2.7**: The number of primes less than is cumulatively approximated by:

.

We conclude that is also a cumulative *non-heuristic* approximation of the number of primes :

**Lemma 2.8**: .

**2C An apparent paradox**

We note that since the above *non-heuristic* estimates appeal to an asymptotic density, and the density of primes tends to as , Figs.1-9 suggest that:

(i) overestimates cumulatively over each of the intervals; whilst

(ii) overestimates over a single cumulative interval.

even though Fig.3 suggests that:

The number of primes in any interval for any given is constant as ; but

The contribution of the expected number of primes in the interval to the total expected number (see Corollary 2.4), , of primes in the interval decreases monotonically.

In other words, an apparent paradox surfaces (as suggested by Fig.9) when we express and the function in terms of the number of primes determined by each function respectively in each interval as follows:

Since as , we can, for any given , define such that:

We conclude that:

**Theorem** For any given , we can find such that the estimated number of primes is always less than the actual number of primes less than plus the number of primes between and as estimated by .

Since (by the Prime Number Theorem), whilst (where ), and for all (see Figs.3 and 4), we note the apparent paradox:

**Corollary** The Prime Number Theorem implies that:

**3 Conventional estimates of for finite are heuristic**

We note that Guy Robin (Robin, G. (1983). “Sur l’ordre maximum de la fonction somme des diviseurs”. Séminaire Delange-Pisot-Poitou, Théorie des nombres (1981-1982). Progress in Mathematics. 38: 233-244.) proved (implicit assumptions?) that the following changes sign infinitely often:

Robin’s result is analogous to *Littlewood’s curious theorem* (implicit assumptions?) that the difference changes sign infinitely often. No analogue of the *Skewes number* (an upper bound on the first natural number for which ) is, however, known for Robin’s result.

Littlewood’s theorem is `curious’ since:

(a) There is no explicitly defined arithmetical formula that, for any , will yield . Hence, Littlewood’s proof deduces the behaviour of for *finite* values of by implicitly appealing to the relation of to , defined as over all integers , through the identity of the infinite summation with the Euler product over all primes, which is valid *only* for Re; as is its consequence (which involves the re-arrangement of an infinite summation that, too, is valid only for Re):

(b) Littlewood’s proof deduces the behaviour of for *finite* values of by (*uncritically, as I argue here?*) appealing to the analytically continued behaviour of in areas where is not defined!

Moreover, we note that—unlike and —*conventional estimates* of for *finite* values of can be treated as *heuristic*, since they appeal only to the *limiting* behaviour (Theorem 420, p.345, Hardy and Wright, Introduction to the Theory of Numbers, 4th ed, 1960, Oxford University Press) of a formally (i.e., explicitly) *undefined* arithmetical function, , to the limiting behaviours of formally *defined* arithmetical functions and :

where:

and, as detailed here, the latter is *curiously* stipulated as valid only for , but definable in terms of a summation over the zeros of the zeta function in the critical strip :

“ is given by the so-called explicit formula

for and not a prime or prime power, and the sum is over all nontrivial zeros of the Riemann zeta function , i.e., those in the critical strip so , and interprets as

."

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