Jacob's ladders and completely new exact synergetic formula for Jacobi's elliptic functions together with Bessel's functions excited by the function $|\zeta(1/2+it)|^2$

TL;DR

This paper introduces new transmutations and synergetic formulas involving Jacobi's elliptic functions, Bessel's functions, and the Riemann zeta function, revealing complex interactions and exact metafunctional equations.

Contribution

It presents five new transmutations of a core formula and ten exact metafunctional equations, advancing the understanding of interactions among classical special functions.

Findings

Five new transmutations of the mother formula

Ten exact metafunctional equations

Revealed synergetic interactions between functions

Abstract

In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set represent a kind of synergetic formulae describing cooperative interactions between corresponding sets of values of basic classical functions.

Equations

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

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

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

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

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

$${\stackrel{(1)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),{\stackrel{(7)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2])\subset \mathbb{C},l=1,2,3$$

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

$$\begin{array}{rl}& |\zeta (s_1^1){|}^2|\mathrm{sn}(s_1^7,k_1){|}^2-\\ & -\{1+\mathcal{O}(\frac{\ln \ln L}{\ln L})\}|\zeta (s_2^1){|}^2|\mathrm{cn}(s_2^7,k_2){|}^2+\\ & +|\zeta (s_3^1){|}^2|\mathrm{dn}(s_3^7,k_3)|=0,\\ & [(k_1{)}^2,(k_2{)}^2,(k_3{)}^2]\in [0,1{]}^3,\end{array}$$

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

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

$$\begin{array}{rl}& {\stackrel{(6)}{\Omega }}_l(U,L,p_l;[|{\zeta }_{0,5}{|}^2]),{\stackrel{(7)}{\Omega }}_l(U,L,k_l;[|{\zeta }_{0,5}{|}^2]),\\ & {\stackrel{(11)}{\Omega }}_l(U,L,p_{l+3};[|{\zeta }_{0,5}{|}^2]),{\stackrel{(12)}{\Omega }}_l(U,L,k_{l+3};[|{\zeta }_{0,5}{|}^2])\subset \mathbb{C},\\ & l=1,2,3\end{array}$$

$$s_l^n=s_l^n([|{\zeta }_{0,5}{|}^2])\in {\stackrel{(n)}{\Omega }}_l,l=1,2,3,n=6,7,11,12$$

$$\begin{array}{rl}& |J_{p_1}(s_1^6){|}^2|\mathrm{cn}(s_2^7,k_2){|}^2|J_{p_4}(s_1^{11}){|}^2|\mathrm{cn}(s_2^{12},k_5){|}^2+\\ & |J_{p_3}(s_3^6){|}^2|\mathrm{cn}(s_2^7,k_2){|}^2|J_{p_6}(s_3^{11}){|}^2|\mathrm{cn}(s_2^{12},k_5){|}^2=\\ & =|J_{p_2}(s_2^6){|}^2|\mathrm{sn}(s_1^7,k_1){|}^2|J_{p_2}(s_2^{11}){|}^2|\mathrm{sn}(s_1^{12},k_4){|}^2+\\ & |J_{p_2}(s_2^6){|}^2|\mathrm{dn}(s_3^7,k_3)||J_{p_5}(s_2^{11}){|}^2|\mathrm{dn}(s_3^{12},k_6){|}^2,\\ & (p_1,p_2,p_3),(p_4,p_5,p_6)\in {\mathbb{Z}}^3,\\ & [(k_1{)}^2,(k_2{)}^2,(k_3{)}^2],[(k_4{)}^2,(k_5{)}^2,(k_6{)}^2]\in (0,1{)}^3.\end{array}$$

$$|\zeta (s_1^1){|}^2,|\zeta (s_2^1){|}^2,\dots ,$$

$${\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2=|{\zeta }_{0,5}{|}^2,$$

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

$$\begin{array}{rl}& \{|J_{p_l}(s)|\},\{|J_{p_{l+3}}(s)|\},l=1,2,3,\\ & \{|\mathrm{sn}(s,k_1)|\},\{|\mathrm{cn}(s,k_2)|\},\{|\mathrm{dn}(s,k_3)|\},\\ & \{|\mathrm{sn}(s,k_4)|\},\{|\mathrm{cn}(s,k_5)|\},\{|\mathrm{dn}(s,k_6)|\},\\ & s\in \mathbb{C}\setminus \{P\},\end{array}$$

$${\stackrel{(8)}{\Omega }}_l={\stackrel{(8)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),l=1,2,3$$

$$\begin{array}{rl}& |\cos s_1^8|=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{1,1}\right)\right|,\\ & |\cos s_2^8|=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{2,1}\right)\right|,\\ & |\cos s_3^8|=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{3,1}\right)\right|;\\ & s_l^8\in {\stackrel{(8)}{\Omega }}_l,\end{array}$$

$${\stackrel{(3)}{\Omega }}_l(U,L),{\stackrel{(8)}{\Omega }}_l(U,L)\subset \mathbb{C},l=1,2,3$$

$$\begin{array}{rl}& |\cos s_1^8{|}^2|\cos s_1^3{|}^2-\left\{1+\mathcal{O}\left(\frac{\ln \ln L}{\ln L}\right)\right\}|\cos s_2^3{|}^2|\cos s_2^8{|}^2+\\ & +|\cos s_3^8{|}^2|\cos s_3^3|=0.\end{array}$$

$$(s{)}^{n_4},(s{)}^{n_5},(s{)}^{n_6},\forall \text{-}(n_4,n_5,n_6)\in {\mathbb{N}}^3.$$

$${\stackrel{(9)}{\Omega }}_l={\stackrel{(9)}{\Omega }}_l(U,L,n_{l+3};[|{\zeta }_{0,5}{|}^2]),l=1,2,3$$

$$\begin{array}{rl}& |s_1^9{|}^{n_4}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{1,1}\right)\right|,\\ & |s_2^9{|}^{n_5}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{2,1}\right)\right|,\\ & |s_3^9{|}^{n_6}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{3,1}\right)\right|,\end{array}$$

$$(n_1,n_2,n_3),(n_4,n_5,n_6)\in {\mathbb{N}}^3$$

$${\stackrel{(4)}{\Omega }}_l(U,L,n_l),{\stackrel{(9)}{\Omega }}_l(U,L,n_{l+3})\subset \mathbb{C},l=1,2,3$$

$$\begin{array}{rl}& |s_1^4{|}^{2n_1}|s_1^9{|}^{2n_4}-\left\{1+\mathcal{O}\left(\frac{\ln \ln L}{\ln L}\right)\right\}|s_2^4{|}^{2n_2}|s_2^9{|}^{2n_5}+\\ & +|s_3^9{|}^{2n_6}|s_3^4{|}^{n_3}=0.\end{array}$$

$${\stackrel{(10)}{\Omega }}_l={\stackrel{(8)}{\Omega }}_l(U,L;[|{\zeta }_{0,5}{|}^2]),l=1,2,3$$

$$\begin{array}{rl}& \frac{1}{|\Gamma (s_1^{10})|}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{1,1}\right)\right|,\\ & \frac{1}{|\Gamma (s_2^{10})|}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{2,1}\right)\right|,\\ & \frac{1}{|\Gamma (s_3^{10})|}=\left|\zeta \left(\frac{1}{2}+i{\alpha }_1^{3,1}\right)\right|,\end{array}$$

$${\stackrel{(5)}{\Omega }}_l(U,L),{\stackrel{(10)}{\Omega }}_l(U,L)\subset \mathbb{C},l=1,2,3$$

Full text

1. Introduction

1.1.

Let us remind that the following set of values

[TABLE]

generate the following secondary asymptotic complete hybrid formula (see [7], (3.7), [6] – [9], $k_{1}=k_{2}=k_{3}=1$)

[TABLE]

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

[TABLE]

and

[TABLE]

is the first reverse iteration of the basic segment

[TABLE]

by means of the Jacob’s ladder, see [2], [3].

1.2.

Next, we have obtained in our paper [8] the set of six transmutations of the mother formula (1.2). For example, there are continuum sets

[TABLE]

such that for every of elements

[TABLE]

we have the following transmutation of (1.2)

[TABLE]

(comp. also the connection between the basic functions in [9], (1.6) – (1.11)).

1.3.

Let us remind that the factor

[TABLE]

which is contained in every element of the set of seven transmutations is the identical one. Consequently, we have used in our paper [9] the operation of crossbreeding (see [4] – [7], i.e. in our case the elimination of the function (1.6)) on every two different elements of the set of transmutations. We have obtained as result the first set of fifteen exact meta-functional equations. For example, (see [9], (3.8)):

[TABLE]

1.4.

In this paper we obtain a set of five new transmutations of the mother formula. Further, by crossbreeding on every two elements of this set we obtain the second set of ten exact meta-functional equations. For example, there are the sets

[TABLE]

such that for every

[TABLE]

we have the following exact meta-functional equation:

[TABLE]

Remark 1*.*

Though the formula (1.10) does not contain the external factors of type (comp. (1.7))

[TABLE]

for example, even so this one is generated (internally) mainly by the function

[TABLE]

(comp. (1.8), (1.9) and [9], (1.6) – (1.11)).

Remark 2*.*

The morphology (external structure for us) of the formula (1.10), as well as for all other formulae in the Theorem, is unambiguous determined by the mother formula (1.2) and by corresponding operation of crossbreeding.

Remark 3*.*

Consequently, the following is true: all ten exact meta-functional equations obtained in this paper are the direct descendants of the mother formula (1.2).

Remark 4*.*

From the point of view of our $\zeta$-alchemy, the global synergetic formula (1.10) represents the result ($\zeta$-compound ) of cooperative interactions excited by the function

[TABLE]

between the sets (substances)

[TABLE]

where $\{P\}$ is the set of corresponding poles. Of course, we may write another nine classes on basis of our Theorem in Section 3 of this paper.

Remark 5*.*

The synergetic formula (1.10) represents the completely new type of formula simultaneously for the following theories:

• (a)

Riemann’s zeta-function,

• (b)

Jacobi’s elliptic functions,

• (c)

Bessel’s functions.

Remark 6*.*

This paper is also based on new notions and methods in the theory of the Riemann’s function we have introduced in our series of 51 papers concerning Jacob’s ladders. These can be found in arXiv[math.CA] starting with the paper [1].

2. Further infinite set of transmutation of the mother formula

2.1.

We define the level curves (comp. [9], (1.10), (1.11))

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L$, correspondingly. Now we have, see (1.2), (2.2) and [8], (4.3), the following

Lemma 1*.*

For every fixed and admissible $U,L$ there are sets

[TABLE]

such that the following formula holds true (transmutation of (1.2)):

[TABLE]

2.2.

Let us consider now the functions

[TABLE]

We define for them the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L,n_{4},n_{5},n_{6}$, correspondingly. Now we have, see (1.2), (2.7) and [8], (4.8), the following

Lemma 2*.*

For every fixed and admissible $U,L$ and for every fixed

[TABLE]

there are the sets

[TABLE]

such that we have the following formula (transmutation of (1.2)):

[TABLE]

2.3.

We define the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L$, correspondingly. Now we have (see (1.2), (2.11) and [8], (5.3)) the following:

Lemma 3*.*

For every fixed and admissible $U,L$ there are sets

[TABLE]

such that the following formula (transmutation of (1.2)) holds true:

[TABLE]

2.4.

We define, for Bessel’s functions

[TABLE]

the level curves

[TABLE]

as the loci

[TABLE]

for fixed and admissible $U,L,p_{4},p_{5},p_{6}$, correspondingly. Now we have (see (1.2), (2.16) and [8], (5.8)) the following:

Lemma 4*.*

For every fixed and admissible $U,L$ and for every fixed

[TABLE]

there are sets

[TABLE]

such that the following formula (transmutation of (1.2)) holds true:

[TABLE]

2.5.

Fot the Jacobi’s elliptic functions

[TABLE]

we define the level curves

[TABLE]

as the loci

[TABLE]

for fixed and admissible $U,L,k_{4},k_{5},k_{6}$, correspondingly. Now we have (see (1.2), (2.21) and [8], (5.13)) the following:

Lemma 5*.*

For every fixed and admissible $U,L$ and for every fixed

[TABLE]

there are sets

[TABLE]

such that the following formula (transmutation of (1.2)) holds true:

[TABLE]

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

We have obtained the following set

[TABLE]

of five further transmutations of the mother formula (1.2). Let us remind that the factor

[TABLE]

(comp. [9], (2.21), (2.22)) which is contained in every element of the set (3.1) is the 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 (3.1).

Remark 7*.*

Let the symbol

[TABLE]

stand for we obtain by crossbreeding of the transmutations of (2.4) and (2.9).

We obtain the following Theorem as the result of crossbreeding on the set (3.1):

Theorem*.*

There are sets (see [9], (1.11))

[TABLE]

and, further, (see (2.1), (2.6), (2.10), (2.15) and (2.20))

[TABLE]

where

[TABLE]

such that for each of the elements

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Remark 8*.*

The following is true: the set of level curves

[TABLE]

is defined as the set

[TABLE]

of loci by means of the values

[TABLE]

(comp. [8], (3.1), (3.2)), where (see [9], (1.10))

[TABLE]

Since

[TABLE]

then we have $c_{l}$’s – as initial conditions – twice determined by the function

[TABLE]

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