Jacob's ladders and wander from Hardy-Littlewood integral (1918) to the $ζ$-Platonic cube form of equivalent of the Fermat-Wiles theorem

Introduction

1.1.

In the paper [12] we have defined the set

of the rectangular signals and we have obtained the following asymptotic formula

on the base of Gramm’s sequence $\{t_\nu\}$.

Remark 1. This formula represents the next sum variant of our almost linear formula, see [10], (25), (27), $r=1$

where the integral in (3) represents new class of increments of the Hardy-Littlewood integral (1918), see [4].

1.2.

Now, let us remind that there are five regular convex polyhedra known as Platonic solids:

(i) regular tetrahedron with four triangular faces,

(ii) regular hexahedron (cube) with six square faces,

(iii) regular octahedron with eight triangular faces,

(iv) regular icosahedron with twenty triangular faces, and

(v) regular dodecahedron with twelve pentagonal faces.

Here, of course, triangular face means equilateral triangle. In this connection see also the nice diagrams in [2], p. 226.

In this paper we obtain new formulas generated by (2) and containing areas of platonic cubes, infinitely small, in $\mathbb{R}^3$.

We have obtained the following main formulas for the set $\{\mathcal{C}_1(n)\}_{n=1}^{N(T)}$ of Platonic cubes $\mathcal{C}_1(n)$: $\zeta$-Platonic cube form:

• of almost linear formula (1.3)

• of the first $\zeta$-functional¹

• of equivalent of the Fermat-Wiles theorem

• of asymptotic decomposition (or expansion)

of the area of Platonic cube with the edge $a=\sqrt{\frac{\pi}{3}}$ by means of areas of $\zeta$-Platonic cubes with infinitely small edges

Remark 2. The $\zeta$-Platonic cube form of the equivalent of Fermat-Wiles theorem represents the point of contact between the Hardy-Littlewood integral (1918) and Fermat-Wiles theorem and also the Platonic hexahedron (cube). The first two notions are separated by the time interval more than 2200 years long from the last notion.

Remark 3. The proof scheme that gives the results (1.4) – (1.7) is adaptable also for the cases (i), (iii) – (v) of the Platonic solids.

Jacob’s ladders: notions and basic geometrical properties

In this paper we use the following notions of our works [7, 8, 9, 10, 11]:

(a) Jacob’s ladder $\varphi_1(T)$,

(b) direct iterations of Jacob’s ladders

for every fixed natural number $k$,

(c) reverse iterations of Jacob’s ladders

where, for example,

for every fixed $k \in\mathbb{N}$ and every sufficiently big $T>0$. We also use the properties of the reverse iterations listed below.

and

Remark 4. The asymptotic behaviour of the points

is as follows: at $T\to\infty$ these points recede unboundedly each from other and all together are receding to infinity. Hence, the set of these points behaves at $T\to\infty$ as one-dimensional Friedmann-Hubble expanding Universe.

2.2. Let us remind that we have proved² the existence of almost linear increments

for the Hardy–Littlewood integral (1918), [4]:

For completeness, we give here some basic geometrical properties related to Jacob’s ladders. These are generated by the sequence

of reverse iterations of the Jacob’s ladders for every sufficiently big $T>0$ and every fixed $k\in\mathbb{N}$.

Property 1. The sequence (2.8) defines a partition of the segment $[T,T^k]$ as follows

on the asymptotically equidistant parts

Property 2. Simultaneously with the Property 1, the sequence (2.8) defines the partition of the integral

into the parts

that are asymptotically equal

It is clear, that (2.10) follows from (2.3) and (2.5) since

while our eq. (2.13) follows from (2.6) and (2.5).

The evolution of the Hardy-Littlewood integral from (1918) up to $\zeta$-Platonic hexahedron form of the Fermat-Wiles theorem

The origin of the Hardy-Littlewood integral lies in the asymptotic formula³

The Littlewood formula, see [5], $0<\delta$ being sufficiently small,

was the next step. And finally, Ingham proved the following formula, see [6],

that contains the complete set of main terms (here $c$ stands for the Euler’s constant).

Remark 5. We see that the first Ingham’s contribution to the formula (23) is the addition of the term $-2cT$ that is missing in previous Littlewood formula (23). The second contribution of Ingham is quite essential improvement of the error term in Littlewood formula, namely the improvement is by 33.3%.

Remark 6. Let us remind also further considerable refinements of the error term, namely

due to Titchmarsh [15] and Balasubramanian [1], correspondingly.

We can conclude previous results into the following statement.

Formula 1.

We will refer to (26) as to Hardy-Littlewood-Ingham formula with the error term $R(T)$.

Further, let us remind the fundamental Good’s $\Omega$-theorem, see [3]:

This statement gives us the exact terminal for the refinements of the type (25).

Remark 7. It is possible to write down

for the error term. In this form $a$ denotes the last valid exponent for which the estimate (28) holds true. Especially, $a=\frac{1}{4}$ represents the Good’s $\Omega$-theorem, and $a=\frac{1}{3}$ represents the main Barasubramanian’s exponent.

Remark 8. It is fully sufficient to use the exponent $\frac{1}{3}$ despite there is also the second Barasubramanian’s exponent $\frac{27}{82}$ that is an improvement by approximately 1.18% of $\frac{1}{3}$, see [1].

Let us remind that we have introduced the Jacob’s ladder

in the paper [7], comp. also [8], where the function $\varphi(T)$ is an arbitrary solution to the non-linear integral equation

where each admissible function $\mu(y)$ generates a solution

We call the function $\varphi_1(T)$ as Jacob’s ladder since the analogy with the Jacob’s dream in Chumash, Bereishis, 28:12.

Remark 9. It is clear that the Jacob’s ladder $\varphi_1(T)$ represents new type of the special function generated by the modulus of the Riemann zeta-function on the critical line, i.e.

For the basic properties of this function see [7, 8, 9].

Next, 83 years after the HLI formula (26), we have proved the following result, see [7].

Formula 2. The Hardy-Littlewood integral has, in addition to previously known HLI expression possessing an unbounded error term at $T\to\infty$, the following infinite number of new and almost exact representations

where $c$ is the Euler constant and $c_0$ is the constant from Titchmarsh-Kober-Atkinson formula.

Further, we have obtained also the following (see [7], (6.2)):

Formula 3.

where the Jacob’s ladder $\varphi_1(T)$ can be seen as the complementary function to the function $(1-c)\pi(T)$ in the sense

Remark 10. In the asymptotic formula (34) we have used the following change⁴

This is completely correct in the asymptotic region since the prime-number law. At the same time, we can give also a motivation for the change (36). Namely, following⁵

we can get that the function $\pi(x)$ is one element of the set of five functions generating the function $\zeta(s)$.

Next, in the paper [10], we have proved the existence of the set of almost linear increments of the Hardy-Littlewood integral

Namely, we have proved the following.

Formula 4. For every fixed $k\in\mathbb{N}$ and every sufficiently big $T>0$ it is true that⁶

$1-c\approx0.42$, $r=1,\ldots,k$, $T\to\infty$,

and⁷

The final step of the proof of (39) contains two main formulae only. Namely, by making use of the substitution $T\to T^{r-1}$ in (26) and $T\to T^r$ in (33) one obtains following formulae:

and

$r=1,\ldots,k$, $T\to\infty$,

correspondingly⁸. If we subtract (41) from (42), we obtain immediately (39).

Remark 11. It is clear that the formula (39) is unreachable without the Jacob’s ladder and its reverse iterations.

Next step in the evolution of the Hardy-Littlewood integral represents the first $\zeta$-functional:

for every fixed $x>0$. This functional was obtained in the paper [11], (4.6).

In the special case of the Fermat’s rationals

we have obtained from (43) also the first point of contact between almost linear increments (39), $r=1$, and the Hardy-Littlewood integral and the Fermat-Wiles theorem. Namely, it is true: the $\zeta$-condition

on the class of all Fermat’s rationals represents the $\zeta$-equivalent of the Fermat-Wiles theorem.

Present contribution to the evolution of the Hardy-Littlewood integral is expressed by the four $\zeta$-Platonic cube formulas (1.4) – (1.7).

Next sum variant of almost linear formula

Let us remind the Riemann real-valued function

where⁹

and, of course,

In present paper we use also the Gramm’ sequence

where the last - Titchmarsh - formula can be found in [14], p. 102. In other words, we use the sequence of roots of the equations $\vartheta(t)=\pi\nu$. Next, we select from the sequence (4.4) a local subsequence

see [12], (3.8) and (3.9), where the letter $n$ denotes the local index, and

We use also the following formulas¹⁰:

correspondingly.

Remark 12. One $\tilde{t}(n) \in[t_n,t_{n+1}]$ is chosen in the mean-value formula (51).

Now, we shall define the partition of $\mathcal{A}_2(n)$ into the set of corresponding squares.

For the segment $[t_n,t_{n+1}]$¹¹ we use the following partition

where

for every fixed $n=1,\ldots,N(T)$.

Now, we give the partition of the segment $[0,Z^2(\tilde{t}(n))]$¹² into the parts of length

(A) First, we shall consider such an $\bar{n}$ that

i.e. the case of bad values of $n$. Since¹³

then¹⁴

Consequently, it is true¹⁵ that

i.e.

Remark 13. The meaning of the previous estimate is that the sum over all bad values $\bar{n}$ gives negligible contribution into the sum (54). Consequently, we can interpret the formula¹⁶

as such one that holds true for the set of good values $n$, i.e. for such $n$ for which it is true that

comp. (58). After re-numbering the good $n$s we can hold the old notation.

(B) We define, in the case (64), the partition of the segment $[0,Z^2(\tilde{t}(n))]$ by the set of the following points

where

and

Consequently¹⁷, it is true

for every fixed $n=1,\ldots,N(T)$, see (64).

4.3.

Now, we define the following set of squares

as the elements of the partition of $\mathcal{A}_2(n)$, where¹⁸

and

for every fixed $n=1,\ldots,N(T)$, see Remark 13, (64).

Next, it follows from (69)¹⁹

Consequently, the following lemma holds true²⁰.

And finally, we have the following result.

Remark 14. The formula (76) expresses the next sum variant of our almost linear formula, see (13), $r=1$.

The $\zeta$-Platonic cube form of equivalent of the Fermat-Wiles theorem

5.1. Let us begin with the next remark.

Remark 15. The subsection 1.2 of this paper contains the set of five existence theorems. For example, by (ii) it is true: the Platonic cube therein is for every $a>0$ as a length of side for every six congruent and generating squares. Simultaneously, $a$ is the edge of the cube.

Further, we introduce the Cartesian coordinate system $O_{tyz}$ in $\mathbb{R}^3$. We use the positive octant

where

Since by (ii) there is the cube $\mathcal{C}[\alpha]$ with the edge²¹

we assign to the set

for every fixed $p$ and $n$ the Platonic cube (its surface)

with the property²²

Consequently, there is, for our case, the complete set

of $\zeta$-Platonic cubes in the set $O^{1}_{tyz}$ in $\mathbb{R}^{3}$. The location of elements (82) over

that is for every fixed $n$, is controlled as follows.

Remark 16. Namely, the location of elements

in corresponding subset fulfils the conditions:

(a)

(b) for the orthogonal projection of every element $\mathcal{C}_{p}(n)$ in the direction $O^{1}_{z}$, i.e.

it is true that

Since the set

contains for every fixed $n=1,\ldots,N(T)$ congruent elements only, we give the new results for the simple case

for example, that is sufficient. In this case we have obtained the following results. First, it is true²³:

Remark 17. The formula (88) expresses the first $\zeta$-Platonic cube form of or almost linear formula, see (13), $r=1$.

Remark 18. From the geometric point of view ($|\mathcal{C}_{1}(n)|$ is area) more exact form of (88) is as follows:

We shall, however, use the form (88).

Now, if we use the substitution

in (88), then we obtain the following result.

Remark 19. The formula (90) represents the $\zeta$-Platonic cube form of the first $\zeta$-functional²⁴.

In the special case of the Fermat’s rationals

we have obtained the following two results.

Next, if we rewrite the eq. (88) in the alternative form²⁵

and remind that²⁶

we obtain the following lemma.

Now, we can introduce the Platonic cube $\mathcal{C}\left[\sqrt{\frac{\pi}{3}}\right]$ with the edge $a=\frac{\pi}{3}\approx1.02$ where, of course,

Consequently, we obtain the following theorem.

I would like to thank Michal Demetrian for his moral support of my study of Jacob’s ladders.

Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynska Dolina M105, 842 48 Bratislava, Slovakia

Email address: [email protected]