Jacob's ladders and exact meta-functional equations on level curves as global quantitative characteristics of synergetic phenomenons excited by the function $|\zeta({1\over2}+it)|^2$

TL;DR

This paper introduces a set of fifteen exact meta-functional equations related to the Riemann zeta-function, derived through crossbreeding of hybrid formulas, offering new insights into its level curves and synergetic phenomena.

Contribution

It presents a novel method of deriving exact meta-functional equations for the Riemann zeta-function using crossbreeding of hybrid formulas, expanding the theoretical framework.

Findings

Fifteen new exact meta-functional equations for the zeta-function

Identification of level curves as global quantitative characteristics

Application of crossbreeding to hybrid formulas

Abstract

In this paper we use operation of crossbreeding on the set of six transmutations of corresponding asymptotic complete hybrid formulas from our previous paper. We obtain in result the set of fifteen exact meta-functional equations. Every of them represents new formula in the theory of the Riemann's zeta-function.

Equations

$$\begin{array}{rl}& \left\{{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2\right\},\{f_1(t)\}=\{{\sin }^{2}t\},\{f_2(t)\}=\{{\cos }^{2}t\},\\ & \{f_3(t)\}=\{\cos 2t\},t\in [\pi L,\pi L+U],U\in (0,\pi /4),L\in \mathbb{N}\end{array}$$

$$\begin{array}{rl}& {\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{1,1}\right)\right|}^2{\sin }^2{\alpha }_0^{1,1}-\\ & -\left\{1+\mathcal{O}\left(\frac{\ln \ln L}{\ln L}\right)\right\}{\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{2,1}\right)\right|}^2{\cos }^2{\alpha }_0^{2,1}+\\ & +{\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{3,1}\right)\right|}^2\cos (2{\alpha }_0^{3,1})=0,\\ & \forall \text{-}L\ge L_0<0,\end{array}$$

$$\begin{array}{rl}& {\alpha }_r^{l,1}={\alpha }_r(U,\pi L,f_l),r=0,1,l=1,2,3,\\ & {\alpha }_0^{l,1}\in (\pi L,\pi L+U),{\alpha }_1^{l,1}\in (\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}),\end{array}$$

$${\stackrel{(1)}{\Omega }}_l,{\stackrel{(5)}{\Omega }}_l,{\stackrel{(7)}{\Omega }}_l(k_l)\subset \mathbb{C},l=1,2,3$$

$$\begin{array}{rl}& \frac{1}{2}+i{\alpha }_1^{l,1}(U,\pi L,f_l)\in {\stackrel{(1)}{\Omega }}_l(U,L),l=1,2,3,\\ & \forall \text{-}U\in (0,\pi /4),\forall \text{-}L\ge L_0>0\end{array}$$

$$s_l^1\in {\stackrel{(1)}{\Omega }}_l,s_l^5\in {\stackrel{(5)}{\Omega }}_l,s_l^7\in {\stackrel{(7)}{\Omega }}_l$$

$$\begin{array}{rl}& \frac{|\zeta (s_1^1){|}^2}{|\Gamma (s_1^5){|}^2}|\mathrm{cn}(s_2^7,k_2){|}^2+\frac{|\zeta (s_3^1){|}^2}{|\Gamma (s_3^5)|}|\mathrm{cn}(s_2^7,k_2){|}^2=\\ & =\frac{|\zeta (s_1^1){|}^2}{|\Gamma (s_2^5){|}^2}|\mathrm{sn}(s_1^7,k_1){|}^2+\frac{|\zeta (s_3^1){|}^2}{|\Gamma (s_2^5){|}^2}|\mathrm{dn}(s_3^7,k_3)|\end{array}$$

$$\begin{array}{rl}& \{|\zeta (s)|\},\{|\Gamma (s)|\},\{\mathrm{sn}(s,k_1)\},\{|\mathrm{cn}(s,k_2)|\},\{|\mathrm{dn}(s,k_3)|\},\\ & [(k_1{)}^2,(k_2{)}^2,(k_3{)}^2]\in (0,1{)}^3.\end{array}$$

$${\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2,{\phi }_1(t),{\phi }_1^1(t),{\alpha }_r^{l,1},r=0,1,l=1,2,3.$$

$${\int }_0^{\mu [x(T)]}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2{e}^{-\frac{2}{x(T)}t}\mathrm{d}t={\int }_0^T{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2\mathrm{d}t,$$

$${\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2⟶{\phi }_1(t),$$

$${\alpha }_r^{l,1}={\phi }_1^{1-r}(d_l),d_l=d(U,\pi L,f_l),r=0,1,l=1,2,3$$

$${\phi }_1(t)⟶{\phi }_1^1(t)$$

$${\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2⟶{\phi }_1(t)⟶{\phi }_1^1(t),$$

$$\begin{array}{rl}& {\alpha }_r^{l,1}={\alpha }_r(U,\pi L,f_l;[|{\zeta }_{0,5}{|}^2]),|{\zeta }_{0,5}{|}^2={\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2,\\ & r=0,1,l=1,2,3.\end{array}$$

$$\begin{array}{rl}& {\stackrel{(1)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),{\stackrel{(2)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),{\stackrel{(3)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),\\ & {\stackrel{(4)}{\Omega }}_l(U,L;n_l,[|{\zeta }_{0,5}{|}^2]),{\stackrel{(5)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),{\stackrel{(6)}{\Omega }}_l(U,L;p_l,[|{\zeta }_{0,5}{|}^2]),\\ & {\stackrel{(7)}{\Omega }}_l(U,L;k_l,[|{\zeta }_{0,5}{|}^2]),l=1,2,3,\end{array}$$

$${\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2$$

$$\begin{array}{rl}& \{f_1(t)\}=\{{\sin }^{2}t\},\{f_2(t)\}=\{{\cos }^{2}t\},\{f_3(t)\}=\{\cos 2t\},\\ & t\in [\pi L,\pi L+U],U\in (0,\pi /4),L\in \mathbb{N}\end{array}$$

$$\begin{array}{rl}& {\tilde{Z}}^2({\alpha }_1^{1,1}){\sin }^2{\alpha }_0^{1,1}+{\tilde{Z}}^2({\alpha }_1^{3,1})\cos (2{\alpha }_0^{3,1})={\tilde{Z}}^2({\alpha }_1^{2,1}){\cos }^2{\alpha }_0^{2,1},\\ & L\ge L_0>0,\end{array}$$

$${\tilde{Z}}^2(t)=\frac{{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2}{\omega (t)},\omega (t)=\left\{1+\mathcal{O}\left(\frac{\ln \ln t}{\ln t}\right)\right\}\ln t,$$

$$\begin{array}{rl}& {\alpha }_r^{l,1}={\alpha }_r(U,\pi L,f_l),r=0,1,l=1,2,3,\\ & {\alpha }_0^{l,1}\in (\pi L,\pi L+U),{\alpha }_1^{l,1}\in (\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}),\end{array}$$

$$[\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}]$$

$$[\pi L,\pi L+U]=[\stackrel{0}{\text{wideparen}\pi L},\stackrel{0}{\text{wideparen}\pi L+U}].$$

$$\Delta (\pi L,U,1)=[\pi L,\pi L+U]\bigcup [\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}]$$

$$\rho \{[\pi L,\pi L+U],[\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}]\}\sim \pi (1-c)\frac{L}{\ln L},L\to \infty ,$$

$$|[\stackrel{r}{\text{wideparen}\pi L},\stackrel{r}{\text{wideparen}\pi L+U}]|=o\left(\frac{L}{\ln L}\right),r=0,1,L\to \infty ,$$

$$|[\pi L+U,\stackrel{1}{\text{wideparen}\pi L}]|\sim \pi (1-c)\frac{L}{\ln L},L\to \infty ,$$

$$\pi L<\stackrel{1}{\text{wideparen}\pi L},U=o\left(\frac{L}{\ln L}\right).$$

$$\omega ({\alpha }_1^{l,1})=\left\{1+{\mathcal{O}}_l\left(\frac{\ln \ln {\alpha }_1^{l,1}}{\ln {\alpha }_1^{l,1}}\right)\right\}\ln {\alpha }_1^{l,1},l=1,2,3.$$

$${\alpha }_1^{l,1}\in (\stackrel{1}{\text{wideparen}\pi L},\stackrel{1}{\text{wideparen}\pi L+U}),$$

Full text

DEDICATED TO 160th ANNIVERSARY OF RIEMANN’s FUNCTIONAL EQUATION.

1. Introduction

1.1.

Let us remind that in the paper [8] we have obtained the following: sets of values

[TABLE]

generates the following secondary asymptotic complete hybrid formula (see [8], (3.7), $k_{1}=k_{2}=k_{3}=1$)

[TABLE]

($L_{0}$ beeing a sufficiently big constant), where

[TABLE]

and the mother formula (1.2) generates the set of six transmutations.

In this paper we use the operation of crossbreeding (see [4] – [7]) on the set of these transmutations to obtain the set of exact meta-functional equations on level curves in the Gauss’ plane. For example, there are sets

[TABLE]

such that they fulfill following initial conditions

[TABLE]

such that for every of elements

[TABLE]

we have the following exact meta-functional equation

[TABLE]

for the corresponding subsets of the following sets

[TABLE]

Remark 1*.*

The exact meta-functional equation (1.4) (together with the set of remaining 14 formulas) represents new type of formulas in the theory of Riemann’s zeta-function as the last level in the sequence:

• (a)

three exact $\zeta$-factorization formulas (see [7], Lemmas 1 –3) ,

• (b)

asymptotic complete hybrid formula,

• (c)

six transmutation of the above,

• (d)

fifteen exact meta-functional equations.

Remark 2*.*

Interpretation of meta-functional equation (1.1) within our $\zeta$-alchemy is that it gives the global quantitative characterization of synergetic phenomenons that lie in the cooperative interactions between corresponding subsets of the set (1.5).

Remark 3*.*

This paper is also based on new notions and methods in the theory of Riemann’s zeta-function we have introduced in our series of 49 papers concerning Jacob’s ladders. These can be found in arXiv[math.CA] starting with the paper [1].

1.2.

Furthermore, we give some remarks concerning connections between basic functions

[TABLE]

Since the function $2\varphi_{1}(T)$ is fixed solution to the integral equation (see [1], comp. [8], subsection 2.1)

[TABLE]

then

[TABLE]

i.e. Jacob’s ladder is generated by the function $\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}$.

Next, we have

[TABLE]

(see [3], (6.3), comp. [8], (2.6)), where

[TABLE]

(of course, the first reverse iteration is generated by the Jacob’s ladder).

Now we have the following connection (see (1.6) – (1.8))

[TABLE]

and, in more details (1.3):

[TABLE]

Remark 4*.*

In our paper [8] we have obtained seven sets of level curves, namely:

[TABLE]

i.e. the main founder of the class of sets (1.11) is the function

[TABLE]

together with three elementary functions in (1.1).

2. The structure of $\mathcal{O}$-terms in (1.1)

2.1.

Let us remind that the sets of values

[TABLE]

generate the following exact secondary complete hybrid formula (see [7], (3.6), $k_{1}=k_{2}=k_{3}=1$)

[TABLE]

(where $L_{0}$ is sufficiently big), where

[TABLE]

(see [2], (9.1), (9.2)). Next, we have

[TABLE]

and

[TABLE]

is the first reverse iteration (by means of the Jacob’s ladder, see [3]) of the basic segment

[TABLE]

Remark 5*.*

The components of the main $\zeta$-disconncted set (for our case)

[TABLE]

are separated each from other by gigantic distance $\rho$, see [3], (5.12), comp. [6], (2.2) – (2.9):

[TABLE]

where $c$ stands for the Euler’s constant.

Remark 6*.*

Disconnected set (2.5) has the following properties (see [3], (2.5 ) – (2.7), $T=\pi L,r=1$):

• (a)

lengths of its components are given by

[TABLE]

• (b)

lengths of adjacent segments are given by

[TABLE]

• (c)

of course,

[TABLE]

2.2.

Next, we have (see (2.2), (2.3))

[TABLE]

Since (see (2.4))

[TABLE]

and (see (2.7) – (2.9))

[TABLE]

then

[TABLE]

and consequently

[TABLE]

Since the function

[TABLE]

is decreasing, then

[TABLE]

Now, we have from (2.10) by (2.12) and (2.13)

[TABLE]

2.3.

Further, we have (see (2.2), (2.14)) the following formula

[TABLE]

and from this, by making use of the following notations

[TABLE]

we obtain

[TABLE]

Since

[TABLE]

then

[TABLE]

and (see (2.14))

[TABLE]

i.e. we have the following

Lemma*.*

The system of sets (2.1) generates the following asymptotic secondary complete hybrid formula

[TABLE]

where

[TABLE]

that is $F$ does not depend on $\alpha_{0}^{2,1}$.

3. List of the first generation of exact meta-functional equations

Let us remind that we have obtained in our paper [8] the set

[TABLE]

of six transmutations of the mother formula (2.21).

Remark 7*.*

It is clear (see Section 2) that the factor

[TABLE]

which is contained in every element of the set (3.1) is identical one.

Consequently, we may apply the operation of crossbreeding (see [4] – [7], here the elimination of the function (3.2)) on every two different elements of the set (3.2).

Remark 8*.*

Let the symbol

[TABLE]

stand for the phrase we obtain by crosbreeding of the transmutations (3.10) and (4.5).

We obtain, as a result of crossbreedings on the set (3.1), the following.

Theorem*.*

There are the sets

[TABLE]

where

[TABLE]

and the sets $\overset{(1)}{\Omega}_{l}$ fulfill the following initial conditions

[TABLE]

(comp. [8], (3.3)) such that for each of the elements

[TABLE]

we have the following set of fifteen exact meta-functional equations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Remark 9*.*

The set of initial conditions (3.3) is an analogue of the Cauchy’s initial conditions for a differential equation. These conditions, apart from other, eliminate some trivial manipulations with the level curves for an usual equations.

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