Jacob's ladders, nonlinear interactions between $\zeta$-oscillating systems and corresponding constraints

TL;DR

This paper introduces a new class of nonlinear interactions between $z$-oscillating systems, revealing asymptotic relationships in a functional algebra framework based on trigonometric functions.

Contribution

It presents a novel class of nonlinear interactions and establishes asymptotic equivalences within a new functional algebra derived from $z$-oscillating systems.

Findings

Cube of two-part form asymptotically equals another two-part form

Main formula generated by subset of trigonometric functions

Introduction of a new class of nonlinear interactions

Abstract

In this paper we introduce new class of nonlinear interactions of $\zeta$-oscillating systems. The main formula is generated by corresponding subset of the set of trigonometric functions. Next, the main formula generates certain set of two-parts forms. For this set the following holds true: the cube of two-part form is asymptotically equal to other two-part form -- short functional algebra.

Equations

$$\prod _{r=1}^k{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r\right)}\right|}^2,k=1,\dots ,k_0,$$

$$(7.1),(7.3),(7.4),(8.2),(8.4),(8.5).$$

$$\begin{array}{rl}& {\cos }^2({\alpha }_0^{2,2})\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^2\right)}\right|}^2+\\ & +{\sin }^2({\alpha }_0^{1,1})\prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^1\right)}\right|}^2\sim 1,\\ & L\to \infty ,1\le k_1,k_2\le k_0,L,k_0\in \mathbb{N},\end{array}$$

$$\begin{array}{rl}& {\alpha }_0^{1,1}={\alpha }_0^1(U,\mu ,L,k_1;{\sin }^{2}t),\dots \\ & {\beta }_r^1={\beta }_r(U,\mu ,L,k_1),\\ & {\alpha }_0^{2,2}={\alpha }_0^2(U,\mu ,L,k_2;{\cos }^{2}t),\dots \\ & {\beta }_r^2={\beta }_r(U,\mu ,L,k_2).\end{array}$$

$${\cos }^{2}t+{\sin }^{2}t=1.$$

$$\begin{array}{rl}& \prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2\sim \\ & \sim \frac{\cos ({\alpha }_0^{2,k_2})}{{\cos }^3({\alpha }_0^{1,k_1})}\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2-\\ & -\frac{U^2}{3}\frac{1}{{\cos }^3({\alpha }_0^{1,k_1}){\cos }^{3}U}\times \\ & \times {\left\{{\cos }^2({\alpha }_0^{3,k_3})\prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2-{\sin }^2({\alpha }_0^{4,k_4})\prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2\right\}}^3,\\ & L\to \infty ,1\le k_1,k_2,k_3,k_4\le k_0,\end{array}$$

$$\int {\cos }^{3}t\mathrm{d}t=\sin t-\frac{1}{3}{\sin }^{3}t,$$

$${\int }_{2\pi L}^{2\pi L+U}{\cos }^{3}t\mathrm{d}t=\sin U-\frac{1}{3}{\sin }^{3}U,L\in \mathbb{N},$$

$$\frac{1}{U}{\int }_{2\pi L}^{2\pi L+U}{\cos }^{3}t\mathrm{d}t=\frac{\sin U}{U}-\frac{U^2}{3}{\left(\frac{\sin U}{U}\right)}^3.$$

$$f_1(t)={\cos }^{3}t\in {\tilde{C}}_0[2\pi L,2\pi L+U],$$

$$t\in [2\pi L,2\pi L+U],U\in \left(0,\frac{\pi }{2}-ϵ\right],$$

$$({\alpha }_0^{1,k_1},{\alpha }_1^{1,k_1},\dots ,{\alpha }_{k_1}^{1,k_1},{\beta }_1^{k_1},\dots ,{\beta }_{k_1}^{k_1}),k_1=1,\dots ,k_0,k_0\in \mathbb{N}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2\sim \\ & \sim \left[\frac{\sin U}{U}-\frac{U^2}{3}{\left(\frac{\sin U}{U}\right)}^3\right]\frac{1}{{\cos }^3({\alpha }_0^{1,k_1})},L\to \infty ,\end{array}$$

$$\begin{array}{rl}& {\alpha }_r^{1,k_1}={\alpha }_r(U,L,k_1;f_1),r=0,1,\dots ,k_1,\\ & {\beta }_r^{k_1}={\beta }_r(U,L,k_1),r=1,\dots ,k_1,\\ & 2\pi L<{\alpha }_0^{1,k_1}<2\pi L+U\Rightarrow 0<{\alpha }_0^{1,k_1}-2\pi L<U,\\ & 1\le k_1\le k_0.\end{array}$$

$$=\left\{1+\mathcal{O}\left(\frac{\ln \ln L}{\ln L}\right)\right\}.$$

$$f_2(t)=\cos t,t\in [2\pi L,2\pi L+U],U\in \left(0,\frac{\pi }{2}-ϵ\right].$$

$$f_2(t)\in {\tilde{C}}_0[2\pi L,2\pi L+U]$$

$$({\alpha }_0^{2,k_2},{\alpha }_1^{2,k_2},\dots ,{\alpha }_{k_2}^{2,k_2},{\beta }_1^{k_2},\dots ,{\beta }_{k_2}^{k_2}),k_2=1,\dots ,k_0,k_0\in \mathbb{N}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2\sim \frac{\sin U}{U}\frac{1}{\cos ({\alpha }_0^{2,k_2})},L\to \infty ,\end{array}$$

$$\begin{array}{rl}& {\alpha }_r^{2,k_2}={\alpha }_r(U,L,k_2;f_2),r=0,1,\dots ,k_2,\\ & {\beta }_r^{k_2}={\beta }_r(U,L,k_2),r=1,\dots ,k_2,\\ & 0<{\alpha }_0^{2,k_2}-2\pi L<U,\\ & 1\le k_2\le k_0.\end{array}$$

$$\begin{array}{rl}& \frac{{\cos }^2({\alpha }_0^{2,2})}{\cos (2\mu +U)}\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^2\right)}\right|}^2-\frac{{\sin }^2({\alpha }_0^{1,1})}{\cos (2\mu +U)}\prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^1\right)}\right|}^2\sim \\ & \sim \frac{\sin U}{U},L\to \infty ,\\ & t\in [\pi L+\mu ,\pi L+\mu +U].\end{array}$$

$$\begin{array}{rl}& \mu =0,L\to 2L,\\ & {\alpha }_r^{2,2}\to {\alpha }_r^{3,k_3},r=0,1,\dots ,k_3,\\ & {\beta }_r^2\to {\beta }_r^{k_3},r=1,\dots ,k_3,\\ & {\alpha }_r^{1,1}\to {\alpha }_r^{4,k_4},r=0,1,\dots ,k_4,\\ & {\beta }_r^1\to {\beta }_r^{k_4},r=1,\dots ,k_4,\end{array}$$

$$t\in [2\pi L,2\pi L+U],U\in \left(0,\frac{\pi }{2}-ϵ\right]$$

$$\begin{array}{rl}& \frac{{\cos }^2({\alpha }_0^{3,k_3})}{\cos U}\prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2-\\ & -\frac{{\sin }^2({\alpha }_0^{4,k_4})}{\cos U}\prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2\sim \frac{\sin U}{U},L\to \infty ,\end{array}$$

$$\begin{array}{rl}& f_3(t)={\cos }^{2}t\to \prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2,\\ & f_4(t)={\sin }^{2}t\to \prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2,\\ & t\in [2\pi L,2\pi L+U],\end{array}$$

$$\begin{array}{rl}& {\alpha }_0^{3,k_3}={\alpha }_0^{2,2}(U,0,2L,k_3;f_3),\dots \\ & {\beta }_r^{k_3}={\beta }_r^2(U,0,2L,k_3),\\ & {\alpha }_0^{4,k_4}={\alpha }_0^{1,1}(U,0,2L,k_4;f_4),\dots \\ & {\beta }_r^1\to {\beta }_r^{k_4},r=1,\dots ,k_4,\\ & {\beta }_r^{k_4}={\beta }_r^1(U,0,2L,k_4).\end{array}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2\sim \\ & \sim \frac{\cos ({\alpha }_0^{2,k_2})}{{\cos }^3({\alpha }_0^{1,k_1})}\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2-\frac{U^2}{3}\frac{1}{{\cos }^2({\alpha }_0^{1,k_1}){\cos }^{3}U}\times \\ & \times {\left\{{\cos }^2({\alpha }_0^{3,k_3})\prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2-{\sin }^2({\alpha }_0^{4,k_4})\prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2\right\}}^3,\\ & L\to \infty ,\\ & 1\le k_1,k_2,k_3,k_3\le k_0.\end{array}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2\sim \\ & \sim \frac{{\cos }^3({\alpha }_0^{1,k_1}}{\cos ({\alpha }_0^{2,k_2})}\prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2+\frac{U^2}{3}\frac{1}{\cos ({\alpha }_0^{2,k_2}){\cos }^{3}U}\times \\ & \times {\left\{{\cos }^2({\alpha }_0^{3,k_3})\prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2-{\sin }^2({\alpha }_0^{4,k_4})\prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2\right\}}^3,\\ & L\to \infty .\end{array}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2\sim \\ & \sim \frac{{\sin }^2({\alpha }_0^{4,k_4})}{{\cos }^2({\alpha }_0^{3,k_3})}\prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2+\\ & +\{\frac{3}{U^2}\frac{{\cos }^{3}U\cos ({\alpha }_0^{2,k_2})}{{\cos }^6({\alpha }_0^{3,k_3})}\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2-\\ & {-\frac{3}{U^2}\frac{{\cos }^{3}U{\cos }^3({\alpha }_0^{1,k_1})}{{\cos }^6({\alpha }_0^{3,k_3})}\prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2\}}^{1/3},\\ & L\to \infty .\end{array}$$

$$\begin{array}{rl}& \prod _{r=1}^{k_4}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{4,k_4}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_4}\right)}\right|}^2\sim \\ & \sim \frac{{\cos }^2({\alpha }_0^{3,k_3})}{{\sin }^2({\alpha }_0^{4,k_4})}\prod _{r=1}^{k_3}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{3,k_3}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_3}\right)}\right|}^2-\\ & -\{\frac{3}{U^2}\frac{{\cos }^{3}U\cos ({\alpha }_0^{2,k_2})}{{\sin }^6({\alpha }_0^{4,k_4})}\prod _{r=1}^{k_2}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{2,k_2}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_2}\right)}\right|}^2-\\ & {-\frac{3}{U^2}\frac{{\cos }^{3}U{\cos }^3({\alpha }_0^{1,k_1})}{{\sin }^6({\alpha }_0^{4,k_4})}\prod _{r=1}^{k_1}{\left|\frac{\zeta \left(\frac{1}{2}+i{\alpha }_r^{1,k_1}\right)}{\zeta \left(\frac{1}{2}+i{\beta }_r^{k_1}\right)}\right|}^2\}}^{1/3},\\ & L\to \infty .\end{array}$$

Full text

1. Introduction

1.1.

Let us remind that we have obtained in our paper [8] certain class of formulae linear in different variables of the following type

[TABLE]

i.e. linear in different oscillating systems. Namely, see formulae in [8]:

[TABLE]

It is, for example, the formula (7.1):

[TABLE]

where

[TABLE]

Remark 1*.*

We have called the formula (1.2) as the $\zeta$-analogue of the elementary trigonometric identity

[TABLE]

Next, we have introduced in our paper [8] the following notions in connection with the formula (1.2):

• (a)

functionally depending $\zeta$-oscillating systems and linearly connected $\zeta$-oscillating systems, (see [8], beginning of section 2.4),

• (b)

interaction between corresponding $\zeta$-oscillating systems (see [8], Definition 4).

Remark 2*.*

By (a) and (b) we may assume in the context of the paper [8] the following:

interaction  =  linear interaction,

(see, for example, (1.3), comp. [9], Remark 1).

Remark 3*.*

Moreover, we notice that the $\zeta$-oscillating system itself (comp. (1.1)) is a complicated nonlinear system (comp. [8], (1.7), i.e. the spectral form of the Riemann-Siegel formula).

1.2.

Next, let us remind the following notions we have introduced (see [1] – [7]) within the theory of the Riemann zeta-function:

• (A)

Jacob’s ladders, (see [1], comp. [2]),

• (B)

$\zeta$-oscillating system, (see [7], (1.1)),

• (C)

factorization formula, (see [5], (4.3) – (4.18), comp. [7], (2.1) – (2.7)),

• (D)

metamorphosis of the $\zeta$-oscillating systems:

• (a)

first, the notion of metamorphoses of an oscillating multiform [4],

• (b)

after that, the notion of metamorphoses of a quotient of two oscillating multiform, [5],

• (E)

$\mathcal{Z}_{\zeta,Q^{2}}$-transformation (or device), [7].

1.3.

In this paper, we shall present certain class of nonlinear interaction formulae, namely:

• (a)

containing nonlinearities in variables of kind (1.1), (comp. Remark 2),

• (b)

describing interaction between corresponding $\zeta$-oscillating systems, i.e. every of these is functionally depending upon others of these $\zeta$-oscillating systems.

The main result is expressed by the following nonlinear formula:

[TABLE]

(comp. (1.3), (1.4) and (a) in the beginning of this section).

2. Lemmas

2.1.

Since

[TABLE]

then

[TABLE]

and

[TABLE]

Of course,

[TABLE]

(the class of functions $\tilde{C}_{0}$ has been defined in our paper [9]), where

[TABLE]

and $\epsilon>0$ is sufficiently small. Consequently, we obtain for generating of the factorization formula by making use our algorithm (see [8], (3.1) – (3.11)) the following

Lemma 1*.*

For the function (2.1) there are vector-valued functions

[TABLE]

($k_{0}$ is arbitrary and fixed) such that the following factorization formula holds true:

[TABLE]

where

[TABLE]

Remark 4*.*

In the asymptotic formula (2.2) the symbol $\sim$ stands for (see [8], (3.8))

[TABLE]

2.2.

Let

[TABLE]

Since

[TABLE]

then we obtain by similar way as in the case of Lemma 1 the following.

Lemma 2*.*

For the function (2.4) there are vector-valued functions

[TABLE]

($k_{0}$ is arbitrary and fixed) such that the following factorization formula holds true:

[TABLE]

where

[TABLE]

2.3.

Next, let us remind (see [8], (8.1) and also (4.2), (4.3), (4.6), (4.7)) the following formula

[TABLE]

Now, if we put (in our present context)

[TABLE]

then we obtain from (2.7) in the case

[TABLE]

the following

Lemma 3*.*

[TABLE]

where

[TABLE]

and

[TABLE]

3. Theorem

3.1.

Now, we obtain by making use of (2.2), (2.5) and (2.7) the following nonlinear interaction formula.

Theorem*.*

[TABLE]

Next, we give following corollaries from the Theorem, (comp. (b) in the beginning of the section 1.3).

Corollary 1*.*

[TABLE]

Corollary 2*.*

[TABLE]

Corollary 3*.*

[TABLE]

3.2.

Let us remind the following correspondences (see (2.1), (2.2); (2.3), (2.4); (2.9)):

[TABLE]

Remark 5*.*

First, we see that the formulae (3.1) – (3.4) define the set of nonlinear interactions of the $(k_{0})^{4}$ oscillating systems in (3.5). Namely, this set contains $4(k_{0})^{4}$ elements of different type. Now, if we use our short notions in (3.5) we may write down the following diagram

[TABLE]

4. Short functional algebra

We shall call each of the following type of functional combinations

[TABLE]

[TABLE]

of two $\zeta$-oscillating systems as the two-parts form. Since

[TABLE]

then the set of formulae (4.1) (just as similar set (4.2)) contains

[TABLE]

two-part forms of different type. In this direction we have (see (3.1)) the following

Corollary 4*.*

[TABLE]

Remark 6*.*

The following property is expressed by the formula (4.3): the cube of two-parts form (4.1) is asymptotically equal to the two-part form (4.2). Of course, we have also the following formula

[TABLE]

Next, it is true (see (4.3)) that for every fixed pair $(k_{3},k_{4})$ we have set of $(k_{0})^{2}$ asymptotic formulae of different types for the cube of corresponding two-parts form (4.1). For example, in the case $k_{0}=10^{3}$ we have $10^{6}$ of these formulae.

Remark 7*.*

Consequently, we may regard the formula (4.3) as a kind of simplification of the well-known school-formula

[TABLE]

in this short functional algebra generated by the formula (3.1). Namely, the right-hand side of (4.3) contains two-parts form (i.e. the two members only in comparison with (4.4)).

5. The iteration formula as a constraint on the corresponding vector-valued functions generated by the operator $\hat{H}$

Let us remind that we have defined (see [8], Definition 2 and Definition 5; [9], Definition) new type of vector-valued operator $\hat{H}$ as follows:

[TABLE]

for every fixed $k$.

Consequently, we have the following set of $k_{0}$ vector-valued functions

[TABLE]

Remark 8*.*

That is, we may say that we have defined the following matrix-valued operator $\hat{H}$:

[TABLE]

Next, it is true that every interaction formula:

• (a)

contains some set of $\zeta$-oscillating systems,

• (b)

every $\zeta$-oscillating system from that set contains the components of corresponding vector-valued function of type (5.1).

Remark 9*.*

Consequently, we may understand every interaction formula (3.1), for example, as the constraint on the set of corresponding vector-valued functions of type (5.1) which are contained in this formula.