Prime numbers with a certain extremal type property

The convex hull of the subgraph of the prime counting function $x\rightarrow \pi(x)$ is a convex set, bounded from above by a graph of some piecewise affine function $x\rightarrow \epsilon(x)$. The vertices of this function form an infinite sequence of points $(e_k,\pi(e_k))_1^{\infty}$. In this paper we present some trivial observation about the sequence $(e_k)_1^{\infty}$ and we formulate a number of questions resulting from the numerical data. Besides we prove one less trivial result: if the Riemann hypothesis is true, then $\lim\frac{e_{k+1}}{e_k}=1$.


Introduction
This paper is a revised and enlarged version of our preprint [7] and of the paper [8] (in Polish). As the preprint [7] attained some interest in the field, we decided to make it published, in spite of the fact that some conjectures stated there were have been in the meantime proved. More precisely, [7] concerns the convex hull of the graph of the function π : [2, ∞) → [1, ∞), which counts the prime numbers in the interval [2, x], and which is usually defined by the formula π(x) = p∈P,p≤x 1 (1) where P = {2, 3, 5, 7, 11, . . .} denotes the set (or the sequence, if necessary) of prime numbers. Some properties related to the graph of the function π were studied in [128] Edward Tutaj 1979 by Carl Pommerance [5] and more recently (2006) by H.L. Montgomery and S. Wagon [6]. In [7], we formulated a number of conjectures concerning the subject and we proved one of them, however assuming the Riemann Hypothesis. Quite recently McNew [2], in his PhD thesis [3], written under the supervision of C. Pommerence, found, applying similar methods as in [7], the proofs of some of the conjectures from [7] without using the Riemann Hypothesis. In the same paper [2], the author discusses the pioneering paper of Pommerence [5] and presents some further results concerning the different types of "extremal" prime numbers.
In this paper we present large parts of the original text of [7]. Some changes are necessary because of the results of McNew, which we will comment in details below and because our set of numerical data is now bigger. The paper of Pommerance [5] was perhaps the first, where the study of convexity of the graph of the function π appeared, but as far as we know, it is in [7] that the sequence (e k ) ∞ 1 of extremal prime numbers appears for the first time as an independent mathematical object. It was noticed by OEIS and lives there under the number A246033. The precise definition of the sequence (e k ) ∞ 1 will be presented in Part 1 of this paper (see also [2]), but at this moment we give an intuitive description. Setting π(1) = 0 we may consider in the plane R 2 the set of points G = {(n, π(n)) : n ∈ N} and its convex hull conv(G). One may say that this convex hull is an unbounded, convex polygon, whose boundary consists of the graph of two continuous functions, namely of the constant function 0 (x) = 0 and of a polygonal function x → (x). The vertices of conv(G) form the sequence (e k , π(e k )) ∞ 0 (where e 0 = 1 and π(e 0 ) = 0) and the sequence (e k ) ∞ 1 is just the sequence of extremal primes. In [7], in [2] and in this paper a number of results and a number of questions are presented, concerning the geometrical structure of conv(G). Before formulating these results, it should be noticed, that this geometrical structure of conv(G) is relatively easy to study numerically. Namely, we consider the bounded polygons G x = conv(G) ∩ ( [1, x] × [0, ∞)) and we count for example its number of vertices π (x) or we calculate the length of its sides. Passing with x to infinity we may study the geometry of conv(G). In [7] and [8] we presented the selected elements of (e k ) for x ≤ 10 12 . In [2] some data are presented for x ≤ 10 13 . In this paper we will present some data for x ≤ 10 17 . The present paper is an example of the papers, where the analysis of a large set of numerical data is a start point to some purely theoretical consideration. Another example worth mentioning is the article of A. Odlyzko, M. Rubinstein and M. Wolf about "jumping champions" [4].
One may ask what is the reason to study the sequence of extremal primes. Well, the function x → (x) is the smallest "reasonable" function bounding the function π(x) from above, but on the other hand, the set (e k ) ∞ 1 of extremal primes is very thin in comparison with the set P of all primes and then one may hope, that it should be easier to tame. Basing on the numerical data we formulated in [7] a number of conjectures. All these conjectures concern the distribution of extremal primes in N. First of them states, that the set of extremal primes is a small subset of P. More exactly The second conjecture states that the set of extremal primes is not to small, since These conjectures are both proved in [2]: Conjecture A is a consequence of Theorem 2.2 there and Conjecture B is stated as Corollary 2.7. Another conjecture speaks about the range of growth at infinity of the function π (x) counting the extremal primes. This was mentioned in [7] and is named in the present paper the γ/2 conjecture. It states -roughly speaking -that the right range of π (x) at infinity is comparable to x γ/2 , where γ is the Euler constant. More exactly, In [2] (Theorem 2.2) it is observed, that β ≤ 2 3 , but this evaluation seems to be far from the best possible. Further discussion of this conjecture will be continued in Part III.
In [7] also the following two conjectures were formulated.
These two last Conjectures are strictly related to each other and they were proved in [7] assuming the Riemann Hypothesis. McNew proves them unconditionally (Corollary 2.8 in [2]). Part I of this paper contains the precise definition of the extremal primes, the presentation of the selected numerical data and the formulation of the number of conjectures. The contents of the present paper is essentially the same as in [7], however there are some differences. In Part II we give the proof of Conjecture D, practically unchanged relative to the original version in [7]. In Part III we show that an important observation from [2] (Theorem 2.4) concerning Conjecture E, can be deduced from the formulas presented in [7].

Definition of extremal prime numbers
In [7] (as well as in [2]) one speaks about the so called extremal prime numbers or convex prime numbers, which may be defined as below. Let us observe first that It is easy to observe that there are many proper subsets F ⊂ P such that It is also not hard to observe that among the sets F ⊂ P having the above property (2), there exists the smallest subset (in consequence only one) E ⊂ P and this is exactly, by definition, the set of extremal primes. Clearly, E is infinite (see Proposition 1) and in many situations it will be more convenient to speak about the strictly increasing sequence E = (e k ) ∞ 1 of extremal prime numbers. It should be noticed here, that in [2] the author considers the convex hull of the set {(n, p n ) : p n ∈ P}, which, clearly, does not make any essential difference. We present below an inductive method of finding e k+1 provided that e k is known.

Proposition 1
The set E is infinite.
Proof. Consider the piecewise affine function : x → (x), whose vertices form exactly the set E. Let l k denote the straight line (the affine function) passing through the points (e k−1 , π(e k−1 )) and (e k , π(e k )). It follows from the definition of extremal points that the graph of the function lies below the line l k . This gives a simple inductive method of finding the next extremal prime e k+1 providing that we know e 1 , e 2 , . . . , e k−1 , e k (in fact it is sufficient to know only e k−1 and e k ). We can do it as follows. We consider the difference quotients of the form for p ∈ P, p > e k . It follows from the remark above, that for each p > e k we have Using the commonly known fact lim p→∞ π(p) p = 0, we have lim p→∞ I k (p) = 0. Then there exists a finite set P k ⊂ P of primes such that q ∈ P k → q > e k and such that I k (p) ≤ I k (q) for p > e k . We set then e k+1 = max P k . This implies that the set E is infinite. Clearly, this means that lim k→∞ e k = +∞.
Prime numbers with a certain extremal type property

[131]
Let us return to the piecewise affine function : x → (x) and consider the sequence i.e. δ n is the slope of the n-th segment lying on the graph of the function . Since the function is strictly increasing and concave, the sequence (δ k ) ∞ 1 is positive and strictly decreasing. Let us observe, that the sequence (δ k ) ∞ 1 may be identified with the derivative of the function . Since δ k is decreasing then the limit δ = lim k→∞ δ k ≥ 0 exists and it must be δ = 0. Indeed, suppose for instance, that δ > 0. Hence for each j ∈ N * we have π(e j+1 ) − π(e j ) > δ(e j+1 − e j ). This implies (adding the above inequalities for 1 ≤ j ≤ k) that for each k ∈ N * the following inequality holds π(e k+1 ) − 1 > δ(e k+1 − 2). But the last inequality is impossible since (once more) lim n→∞ π(n) n = 0. The property δ = lim k→∞ δ k = 0 makes it possible to observe that the set P \ E is infinite.
Proof. This is almost obvious from the intuitive point of view. However, a short proof we present here is related to the very non-trivial results about small gaps between primes. Suppose, for the sake of contradiction, that P \ E is finite. Hence for sufficiently great a > 0 we have E ∩ [a, ∞) = P ∩ [a, ∞). In consequence we have where p n ∈ P. This would imply, that lim n→∞ (p n+1 − p n ) = ∞. But this is impossible, because we know now from many recent results (for example of Zhang, [9]), that lim inf(p n+1 − p n ) < 7 · 10 7 . Since the paper of Zhang the constant 7 · 10 7 was considerably diminished.
The observations about the extremal primes made above are rather elementary. We will speak later about some deeper results. The problem with the sequence of extremal primes is in some sense similar to the problem we have with sequence of all primes and with its subsequences like for example the sequence (conjectured infinite) of twin primes. Namely, it is relatively easy to produce the consecutive elements of the sequence (e k ) ∞ 1 but it is rather hopeless to find an (exact) analytical formula describing the set of extremal primes. Now, it is perhaps a good moment to notice, that it is practically impossible to calculate "by hand" the elements of the sequence E. Using Proposition 1 we may find ten or twenty first terms of the sequence (e k ) ∞ 1 without using computers, but for to go further we need strong calculating machines.
[132] Edward Tutaj We have calculated more than the first 50000 extremal primes and after studying these numerical data, we can formulate a number of more or less interesting questions. It is impossible to give here the complete list of the first 50000 extremal primes, but we present below some selected data. The list of e k , where k ≤ 3000 and k ≡ 0(mod 100) and the list of e k , where k ≤ 50000 and k ≡ 0(mod 10000). The examination of the sequence of the first 50000 extremal primes allows us to formulate a number of questions. First of all it seems to be interesting to say something about the density of the sequence (e k ) ∞ 1 . Our experimental data support some conjectures. Namely, It follows from our data that Let us remember that, as it was mentioned in Introduction, this two conjectures are proved in [2].
Since the set E of extremal prime numbers is infinite and, clearly, the problem of finding any reasonable explicit formula describing the correspondence N n → e n is rather out of reach, we will define and try to study a function, which may be called extremal primes counting function π . The formula for π is analogous to formula (1). We set Unfortunately we know only 50000 values of π (x) for x ≤ 10 17 . However, it seems to be possible to formulate some conjectures about π . Clearly, π e (x) ≤ π(x) and the growth of π is much slower than the growth of π. For example, π (x 0 ) = 1700, when x 0 = 196062395777 and for the same x 0 we have π(x 0 ) = 7855721212. In particular, we may try to find the best α < 1 such that π (x) = o(x α ) observing the ratio ln(n) ln(en) when n tends to infinity (in our case only for n ≤ 10 17 ). Maybe only accidentally, but the best α obtained from our data is near to γ 2 , where γ is the Euler constant. Hence we formulate.
Conjecture H: (see [7], Conjecture C in Introduction, also [2]) There exists infimum Our numerical data support strongly also the following interesting conjecture.
We will prove below, in Part II, that the Riemann Hypothesis implies Conjecture I. This conjecture is interesting itself, but also because of the following observation.
where p n ∈ P.
Proof. For each n ∈ N there exists k(n) ∈ N such that and the last sequence tends by our assumption to 1. Let us recall here, that lim n→∞ pn+1 pn = 1 implies Prime Number Theorem. This was proved by P. Erdös in his elementary proof of PNT (see [1]).
Prime numbers with a certain extremal type property
Another phenomenon is related to the inequality I k (p) ≤ I k (p o ), which is described in Proposition 1. One may ask if the number of points p > e k such that I k (p) = I k (p o ) is greater than 1. In our numerical data we have only three such examples, namely for k = 2 we have I 2 (5) = I 2 (7) , I 3 (13) = I 3 (19) and also I 4 (23) = I 4 (31) = I 4 (43) = I 4 (47) = 1 4 = δ 4 but in fact our programme searching next extremal primes was not written to search such exceptions.

Part II
As we announced in Introduction in this Part we present the original proof of Conjecture I (from [7]) depending on Riemann Hypothesis.

Definition of lenses
The gaps between extremal primes will be called lenses. More exactly, Definition 5 Given a positive integer k ∈ N the lens S k is a set The difference e k+1 − e k will be called the length of the lens S k and will be denoted by |S k |.
Since we will apply in the sequel the language of differential calculus, it will be more comfortable to work with the function [ We consider the following -well known -functions L and ε called integral logarithm and error term respectively, defined by the following formulae: and Together with L and ε we consider the function and for x ∈ (2, ∞) and h ∈ R, [136]

Edward Tutaj
Clearly, all these functions are analytic at least in (2, ∞). We will use the derivatives of the considered functions to the order four and we shall write y instead of ln x to present some formulas in more compact form. Hence we have and further derivatives The derivatives of error term function, written in an analogous manner, run as follows Let us observe, that the second derivatives of the functions L and ε are negative, so both these functions are concave. The second derivative of the function ϕ has the form then taking into account that There exists x 0 ∈ (2, ∞) such that the function ϕ is concave in the interval [x 0 , ∞).

A remark on Taylor polynomials of considered functions
Let us fix a point x ∈ (2, ∞). Let T x,L denote the Taylor polynomial of order three of the function L with the center at x. Hence Prime numbers with a certain extremal type property

[137]
The remainder R x,L (h), written in the Lagrange form, is given by the formula where ξ is a point from the (x, x + h). Since L (4) < 0 in all its domain, we have the inequality.
Let T x,ε denote the Taylor polynomial of order three of the function ε with the center at x, i.e.
Using an analogous argumentation as in the case of the function L we have

Definition of two functions
In this section we will define two functions h + : where x 0 is the point defined in Proposition 6. First we will describe in details the definition of the function h + . The definition of h − will be similar.
Let us fix a point x ∈ (x 0 , ∞). Take into account the tangent line l(x, h) to the graph of the function ϕ at the point (x, ϕ(x)). Its equation for h ∈ R is given by The tangent half-lines obtained, when we restrict ourselves in the formula (10) to h ∈ [0, ∞) or h ∈ (−∞, 0] will be denoted by l + (x, h) or l − (x, h), respectively. For h = 0 we have the inequality This means that the half-line l + starts from the interior point (x, ε(x)) of the subgraph of the function L + ϕ, which is a convex set. Since On the other hand, hence the half-line l + (x, h) must intersect the graph of the strictly concave function L(x + h) + ε(x + h) in exactly one point. Hence we have proved the following

Proposition 9
For each x ∈ (x 0 , ∞) there exists exactly one positive number h + (x) such that In other words, for each x ∈ (x 0 , ∞) the equation (with unknown h) has exactly one positive solution, which we will denote by h + (x). If one replaces the half-line l + (x, h), by the half line l − (x, h), then applying the same arguments as above, we obtain

Proposition 10
For each x ∈ (x 0 , ∞) there exists exactly one negative number h − (x) such that In other words, equation (11) has exactly one negative solution, which we will denote by h − (x).

An auxiliary equation
In this paper we would like to establish the order of magnitude of the functions , when x tends to +∞). Since the equation (11) is rather hard to solve, we will consider an auxiliary equation T which can be written in the form Equation (13) is an algebraic equation of degree three. It has at least one real root. We will see that it can have (and has) more then one real root and we will be interested not only in the existence of roots of equation (13), but also on theirs Prime numbers with a certain extremal type property signs. Let us observe, that since W x (0) = 2ε(x) > 0, the number h = 0 cannot be a root of considered equation. Let us also observe that, in fact, equation (13) is not a single algebraic equation, but it is a one parameter family of algebraic equations, where the parameter is x ∈ (x 0 , ∞). We will prove the following result.
Lemma 11 (i) There exists x + ∈ (x 0 , ∞), such that for each x > x + the equation W x (h) = 0 has a positive root.
The proof of the lemma is done together with the proof of Proposition 16. Assume now that Lemma 11 is true. This allows us to define two new functions h * + and h * − . We will describe in details the definition of h * + . We set Definition 12 Let x ∈ (x + , ∞). Then the set of positive roots of Equation (13) is not empty and we set The relation between the functions h + and h * + is the following x,ε + T (3) x,L because of inequality (9). Hence the equation W x (h) = 0 has no roots in the interval h ∈ [0, h + (x)]. But this means that h + (x) < h * + (x), which ends the proof of Proposition 13.
Assume once more that Lemma 11 is true. We have ∞). Then the set of negative roots 13 is not empty and we set The relation between the functions h − and h * − is as follows

Proposition 15
If Lemma 11 is true, then for The proof of Proposition 15 is similar to the proof of Proposition 13, so we skip it. [140]

The proof of the main result
Now we will prove the Proposition 16 formulated below. Equation (13) we are interested in, can be written in the form where, using formulas (5)-(13) we have Now, taking into account the fact, that for x sufficiently large A 3 (x) > 0, we divide equation (14) by A 3 (x) in order to obtain the form where For further analysis of (15) it will be convenient to use some Landau symbols. Let us recall that for a function g defined in the neighbourhood of +∞ one writes g = o (1) if and only if lim x→+∞ g(x) = 0. Using this convention, we can write (1) , This makes it possible to write 15 in the form Prime numbers with a certain extremal type property

[141]
Now apply the substitution h = θx, which leads to the form Since we work only with x > 0, we can divide the last equation by x 3 , and obtain the following equation (with unknown θ), Finally, taking into account the equality we can write equation (16) in the form where v 1 (x), v 2 (x), v 0 (x) are three positive functions defined in a neighbourhood of +∞ and tending to 0 when x tends to +∞. If for a fixed x we find a number θ being a root of equation (17), then the number h = θ · x is a root of 15. It is then enough to study equation (17). We shall prove much more. Namely we have the following result.
Proof. Indeed, Proposition 16 is stronger than Lemma 11, where we need only the existence of a negative root and of a positive root. In Proposition 16 we prove not only that the roots exist, but also that we can find the solutions in an arbitrary open interval containing the origin. Without loss of generality, we may assume, that α ≤ 1. Let us fix then a positive number 1 ≥ α > 0 and choose x 2 so large, that for and Such an x 2 exists since all three functions v 2 , v 1 , v 0 are o(1) when x tends to +∞. Let us fix x > x 2 . We rewrite equation (17)) in the form f (θ) = g(θ), where and g(θ) = 3 · θ 2 . [142]

The order of magnitude of lenses
By the results of the previous subsection, we can consider four functions h − , h + , h * − and h * + , which are defined in an interval (M, ∞), and such that the following inequalities hold for each x ∈ (M, ∞), Our aim is to establish the order of magnitude at +∞ of the difference H(x) = h + (x) − h − (x). We will prove the following result.

Proposition 17
The function H satisfies the relation when x tends to +∞.
Proof. This follows directly from the property formulated in Proposition 16. Indeed, it is sufficient to show separately that h To prove the first relation, let us fix a positive number > 0. It follows from Proposition 16 (setting α = ) that there exists M 1 > M , such that x > M 1 implies, that there exists a number θ < (θ depending on x) such that h * Now we can prove a theorem on the order of magnitude of the length of lenses S k using Proposition 17. First we shall prove the following lemma about sequences tending to +∞.

Prime numbers with a certain extremal type property [143]
Lemma 18 Suppose that we have four sequences lim k→∞ e k = +∞, Then lim k→∞ e k+1 − e k e k = 0.
Proof. From (21) and (23) we deduce that It must be also lim Indeed, suppose that there exists an infinite subset L ⊂ N and a constant K > 0 such that 0 ≤ x − n ≤ K for n ∈ L. Then for n ∈ L we have Hence by (24), This implies that lim n∈L z n = +∞. In consequence, lim n∈L x + n z n = 0, thus there exists n ∈ L such that x + n < z n , but this is impossible. From the inequality and this ends the proof of Lemma 18.

Lemma 19
The graph of the function π * lies between the graphs of the functions L − ε and L + ε, where the functions L and ε are defined by (3) and (4).
Proof. Suppose the opposite. Then there exist two consecutive prime numbers p n and p n+1 such that the points A = (p n , n) and B = (p n+1 , n + 1) lie between L − ε and L + ε and the segment [A; B] cuts the graph of L − ε or L + ε. But the subgraph of L + ε is convex, then [A; B] cuts only the graph of L − ε. This means, that there exists a point x ∈ (p n , p n+1 ) such that the point X = (x, n) lies below the graph of L − ε. But X = (x, π(x)), then from the definition of the error term, X lies between the graphs of L − ε and L + ε. This ends the proof of Lemma 19.
Lemma 20 Let S k be a lens defined by the extremal prime numbers e k and e k+1 . Then the straight line joining the points U = (e k , π(e k )) and V = (e k+1 , π(e k+1 )) cannot cut the graph of L − ε in two distinct points.
Proof. This follows from the Lemma 19 since, by the definition of extremal points, the whole graph of π * lies below the straight line joining the points U and V .
The main theorem of this section is the following.

Theorem 21
With the notations as above if the Riemann Conjecture is true, then lim k→+∞ e k+1 e k = 1.
Proof. Let U and V be as in Lemma 20. Take the straight line l(U, V ) joining U and V and translate it to the position l * , where the straight line l * is parallel to l(U, V ) and tangent to the graph of L − ε. This line l * cuts the graph of L + ε in points U * and V * , whose first coordinates are x − k and x + k respectively, and the tangent point is z k . It is not hard to check, that the sequences ( and (e k ) ∞ 1 satisfy the assumptions of Lemma 18. Then this ends the proof of the theorem.
We have an equivalent formulation.

Corollary 22
The length of lenses x → S(x) satisfies the equality S(x) = o(x).

Prime numbers with a certain extremal type property
[145]

Additional remarks
In [7] we wrote: It is natural to ask if one can prove the results like Theorem 21 or Corollary 22 without assuming the Riemann Hypothesis. Maybe this is possible, but it seems, that the method used in this paper is insufficient. And also: I was not able to prove Theorem 21 using L(x) = Li(x) and A(ln x) 3 5 (ln(ln x)) 1 5 .
(Un)fortunately it appeared, that we were to pessimistic. McNew in [2], using similar methods as in [7], but applied to the Vinogradow error term (25), proved Conjecture I without assuming the Riemann Hypothesis. In the same paper [2], he gave unconditional proofs of Conjectures F and G.

The conjecture γ/2
Among the conjectures formulated in [7], the Conjecture H seems to be the most interesting. It is related to the stronger version of the Corollary 22, which is proved in the present paper with the use of the Riemann Hypothesis, but which is true, as it was proved in [2], unconditionally. In the same paper [2] it is observed, that the Riemann Hypothesis allows us to formulate a stronger version of Corollary 22. McNew deduces this version from his proof of the theorem on the behaviour of the sequence (e k ) ∞ 1 of extremal primes. We will check below, that a little deeper analysis of the proof of Corollary 22 given in the present paper, leads to the same conclusion as Theorem 2.4 in [2]. Namely, we have the following.

Theorem 23
In the notation as in the previous section, there exists a constant C > 0 such that where y = ln(x).
Before proving this theorem, we will return to the proof of the relation S(x) = o(x). As we have observed, the function S(x) is controlled by the function H(x) = h + (x) − h − (x) considered in Proposition 17. Hence to control the function H(x) it is sufficient to control the function x → h * + (x) defined by the relation h * + (x) = θ(x) · x, where the function θ(x) satisfies the implicit equation (16)) We have proved above, that this implicit equation has a positive solution x → θ(x), such that lim x→∞ θ(x) = 0 and this was enough for S(x) = o(x). First we will prove a proposition, which is weaker than Theorem 23 but stronger than S(x) = o(x).
[146] Edward Tutaj Proposition 24 For each k ∈ N there exists M k > 0 such that Proof. We will use the inductive argument. Taking into account the particular form of the coefficients v 2 (x), v 1 (x) and v 0 (x) we may state, that there exist the polynomials U 2 (y), U 1 (y) and U 0 (y) (with respect to y = ln(x)) such that the following inequality holds Clearly, we may assume, that 0 < θ < 1 (for x sufficiently large), and in consequence, that θ 3 < θ 2 . Hence, we may deduce from (28) that where U (y) is a polynomial with respect to y = ln(x). Hence, for x large enough, we have the inequality which gives the inequality from (27) for k = 1. Assume now, that there exists M k > 0 such that for x > M k we have We will prove that there exists M k+1 > 0 such that for x > M k+1 we have We return once more to inequality (28). Using θ < 1 we can obtain from this inequality that where U 3 (y) is a polynomial with respect to y = ln(x). Hence, for sufficiently large x > M k+1 , we have the inequality Prime numbers with a certain extremal type property

[147]
Since, clearly we may assume that M k+1 > M k then it follows from the inductive assumption, that for x > M k+1 we have 2θ 2 < 6 y 3 k y k , and thus This ends the proof of Proposition 24.
It appears, that one can "squeeze" much more from equation (26) in order to obtain the proof of Theorem 23 presented below.
Proof of Theorem 23. We return to inequality (28). Namely, To obtain Proposition 24 we used only the fact that the functions U 2 (y), U 1 (y) and U 0 (y) are polynomials. Clearly, one may find many polynomials, U 2 , U 1 , U 0 for which inequality (28) holds. In particular, one may choose the above polynomials to have all the degree 3 and not to big coefficients. More exactly, there exists a constant M > 0 such that for x > M we get θ(x) < 1 and 3θ 2 < θ 3 + 6 y + 2 Taking into account the fact that θ < 1 and θ 3 < θ 2 we obtain the inequality which may be rewritten in the form Finally, we choose a constant K > M such that y y−1 < 5 4 and we obtain the inequality which is valid for x > K. Clearly, it is enough to prove the Theorem 23 since h * + (x) = θ(x) · x.
Theorem 23 brings us some information about Conjecture H. Since the sum of lenses contained in the interval [2, x] equals -roughly speakingx and their number is of order x α , then x α · x 3 4 must be as large as x. Hence α ≥ 1 4 . This observation is mentioned in [2] (Corollary 2.6). On the other hand, there is a strong numerical argument supporting the conjecture α = γ/2. We will present these numerical data in a future paper, however, in the next subsection, we give some tables and some graphs for to illustrate what we mean by the term "supporting argument".

Some more numerical data
At present we know the exact values of the sequence e k for k ≤ 8 · 10 4 . The tables inserted below contain some selected data, which one may use to confirm (or -if one prefers -to disprove) the γ 2 conjecture formulated as follows. Let us denote β k = ln(k) ln(e k ) . Then the γ/2 conjecture say simply that lim β k = γ/2. The presented data seem to be promising with respect to the γ/2 conjecture. On the other hand, the examples like the conjecture of Mertens, show that one should be careful. The fact, that the constant γ is present in many theorems of analytical theory of numbers is, perhaps, an additional argument for optimists.
Below we present also two pictures. First of them shows the convex hull of the graph of the function π(x) for 1 ≤ x ≤ 113. The second shows the behaviour of the ratio π e (x)/x