Jacob's ladders and infinite set of transmutations of asymptotic complete hybrid formula on level curves in Gauss' plane

TL;DR

This paper introduces a novel phenomenon where a fixed asymptotic hybrid formula generates infinite transmutations linking the Riemann zeta function's values with moduli of integral and meromorphic functions, revealing new mathematical relationships.

Contribution

It presents the discovery that each 'mother' hybrid formula produces an infinite set of transmutations connecting zeta function values with other complex functions.

Findings

Infinite transmutations of hybrid formulas are generated from a single mother formula.

Each transmutation links subsets of |zeta(s)| with moduli of integral and meromorphic functions.

New relationships in the complex plane are established through these formulas.

Abstract

In this paper we have obtained new phenomenon lying in the following: every fixed asymptotic complete hybrid formula (we call it as mother formula) generates infinite set of new formulas (transmutations) such that every new formula expresses a close binding between some subset of $\{|\zeta(s)|\}$ and subset of moduli of certain integral and meromorphic functions.

Equations

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

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

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

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

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

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

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

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

$$s_l^1\in {\stackrel{(1)}{\Omega }}_l,s_l^2\in {\stackrel{(2)}{\Omega }}_l$$

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

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

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

$$\frac{|\zeta (s_1^1){|}^2}{|\Gamma (s_1^5){|}^2}-\left\{1+\mathcal{O}\left(\frac{\ln \ln L}{\ln L}\right)\right\}\frac{|\zeta (s_2^1){|}^2}{|\Gamma (s_2^5){|}^2}+\frac{|\zeta (s_3^1){|}^2}{|\Gamma (s_3^5)|}=0,$$

$${\stackrel{(7)}{\Omega }}_l(U,L,k_l)\subset \mathbb{C},l=1,2,3,[(k_1{)}^2,(k_2{)}^2,(k_3{)}^2]\in (0,1{)}^3$$

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

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

$${\phi }_1(t)=\frac{1}{2}\phi (t)$$

$${\int }_0^{\mu [x(T)]}{Z}^2(t)e^{-\frac{2}{x(T)}t}\mathrm{d}t={\int }_0^T{Z}^2(t)\mathrm{d}t,$$

$$\begin{array}{rl}& Z(t)=e^{i\vartheta (t)}\zeta \left(\frac{1}{2}+it\right),\\ & \vartheta (t)=-\frac{t}{2}\ln \pi +\mathrm{Im}\left\{\ln \Gamma \left(\frac{1}{4}+i\frac{t}{2}\right)\right\},\end{array}$$

$$y=\phi (T;\mu )=\phi (T);\mu (y)>7y\ln y.$$

$${\int }_0^T\left|\zeta \left(\frac{1}{2}+it\right)\right|\mathrm{d}t$$

$${\int }_0^T\left|\zeta \left(\frac{1}{2}+it\right)\right|\mathrm{d}t={\phi }_1(T)\ln \{{\phi }_1(T)\}+(c-\ln 2\pi ){\phi }_1(T)+c_0+\mathcal{O}\left(\frac{\ln T}{T}\right)$$

$$T\to \infty ,$$

$$T-{\phi }_1(T)\sim (1-c)\pi (T);\pi (T)\sim \frac{T}{\ln T},T\to \infty ,$$

$$(1-c)\pi (T)$$

$${\phi }_1(T)+(1-c)\pi (T)\sim T,T\to \infty .$$

$$\{\stackrel{k}{T}{\}}_{k=1}^{k_0},k_0\in \mathbb{N}:{\phi }_1(\stackrel{k}{T})=\stackrel{k-1}{T},\stackrel{0}{T}=T>T_0>0,\stackrel{k}{T}={\phi }_1^{-k}(T),$$

$$[T,T+U],U=o\left(\frac{T}{\ln T}\right),T\to \infty$$

$$\Delta (T,U,k)=\bigcup _{r=0}^k[\stackrel{r}{\text{wideparen}T},\stackrel{r}{\text{wideparen}T+U}],1\le k\le k_0$$

$$|[\stackrel{r}{\text{wideparen}T},\stackrel{r}{\text{wideparen}T+U}]|\sim o\left(\frac{T}{\ln T}\right),T\to \infty ,$$

Full text

Dedicated to old alchemists

1. Introduction

1.1.

Let us remind that the following sets of values

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

is the first reverse iteration (by means of the Jacob’s ladder, see [4]) of the basic segment

[TABLE]

Remark 1*.*

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

[TABLE]

are separated each from other by gigantic distance $\rho$, (see [4], (5.12), comp. [7], (2.2) – (2.9))):

[TABLE]

($c$ stands for the Euler’s constant).

1.2.

In this paper we obtain, for example, the following transmutations of the mother formula (1.2). For every fixed and admissible $U$ and $L$ (see (1.1), (1.2)), the formula (1.2) generates:

• (a)

continuum sets

[TABLE]

such that for any elements

[TABLE]

we have the following formula (transmutation of (1.2))

[TABLE]

• (b)

the continuum set

[TABLE]

such that for any elements

[TABLE]

we have the following formula (transmutation of (1.2))

[TABLE]

• (c)

the continuum set

[TABLE]

(for every fixed and admissible $k_{1},k_{2},k_{3}$) such that for any fixed elements

[TABLE]

we have the following transmutation of (1.2)

[TABLE]

(for Jacobi’s elliptic functions of moduli $k_{1},k_{2},k_{3}$).

1.3.

Now we give the following remarks.

Remark 2*.*

The formula (1.1) gives the example of the continuum set of transmutations of the mother formula (1.2).

Remark 3*.*

Let us remind that the mother formula is written as orthogonality condition for corresponding vectors. We see that the transmutations (1.6), (1.8) as well as (1.10) inherit this property.

Remark 4*.*

The method presented in this paper is general one in the following sense:

• (a)

single secondary asymptotic complete hybrid formula (1.2) generates the infinite set of its transmutations,

• (b)

there is an infinite set of asymptotic complete hybrid formulas, (see [5] – [8] concerning evolution from $\zeta$-factorization formula to complete hybrid formula).

Remark 5*.*

Let us remind that the general feature of all transmutations is a new phenomenon that lies in close binding between some subsets of $|\zeta(s)|$ and subsets of moduli of other integral and meromorphic functions.

Remark 6*.*

This paper is based also on new notions and methods in the theory of the Riemann zeta-function we have introduced in our series of 48 papers concerning Jacob’s ladders. These can be found in arXiv [math.CA] starting with the paper [2].

2. Jacob’s ladder as basis of secondary complete hybrid formula

2.1.

Let us remind that the Jacob’s ladder

[TABLE]

was introduced in [2], (see also [3]), where the function $\varphi_{1}(t)$ is arbitrary solution of the non-linear integral equation (also introduced in [2])

[TABLE]

of course,

[TABLE]

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

[TABLE]

We call the function $\varphi_{1}(T)$ the Jacob’s ladder as an analogue of the Jacob’s dream in Chumash, Bereishis, 28:12.

Remark 7*.*

By making use of the Jacob’s ladders we have shown (see [2]) that the classical Hardy-Littlewood integral (1918)

[TABLE]

has - in addition to previously known Hardy-Littlewood expression (and other similar) possessing an unbounded error term at $T\to\infty$ - the following infinite set of almost exact representations

[TABLE]

as

[TABLE]

where $c$ is the Euler’s constant and $c_{0}$ is the constant from the Titschmarsh-Kober-Atkinson formula.

2.2.

Next, we have obtained (see [2], (6.2)) the following formula

[TABLE]

where $\pi(T)$ is the prime-counting function.

Remark 8*.*

Consequently, the Jacob’s ladder $\varphi_{1}(T)$ can be viewed by our formula (2.1) as an asymptotic complementary function to the function

[TABLE]

in the following sense

[TABLE]

Since Jacob’s ladder is exactly increasing function we have the reversely iterated sequence

[TABLE]

where $T_{0}$ is a sufficiently big (we fix $k_{0}$). Next we have (see [4], (2.5) – (2.7), (5.1) – (5.13)) the following basic property generated by the Jacob’s ladder.

Property*.*

For every segment

[TABLE]

there is the following class of disconnected sets

[TABLE]

generated by the Jacob’s ladder $\varphi_{1}(t)$.

Remark 9*.*

Disconnected set (2.2) has the following properties (see [4], (2.5) – (2.7)):

• (a)

lengths of its components are given by

[TABLE]

• (b)

lengths of adjacent segments are given by

[TABLE]

i.e. for distance $\rho$ of two consecutive components we have

[TABLE]

Remark 10*.*

Formula (1.4) follows from (2.5) for $r=1$ and $T=\pi L$.

Remark 11*.*

Asymptotic behavior of the components of the disconnected set (2.2) is as follows: at $T\to\infty$ these components recede unboundedly each from other and all together are receding to infinity. Hence, at $T\to\infty$ the set of the components of (2.2) behaves as one-dimensional Friedman-Hubble universe.

2.3.

Finally, let us remind that the functions (see (1.2), (1.3))

[TABLE]

are generated by the Jacob’s ladder $\varphi_{1}(T)$ and its iterations as follows (see [4], (6.3), [5], (1.14), [7], (3.6))

[TABLE]

3. Transmutation of type $[\zeta,\zeta]$

3.1.

Let us remind that the loci

[TABLE]

is the level curve, (comp. [9], pp. 120 – 122). Of course, the topological structure of the level curve is (in general) very complicated. For example, the level curve

[TABLE]

contains three ovals (apart from another) situated in the neighborhood of the first three roots of the equation

[TABLE]

i.e. this one represents the disconnected set, (see [1], p. 371, FIG. 149).

3.2.

First, we define for every admissible and fixed $U,L$ (see (1.1), (1.2)) the following level curves:

[TABLE]

as the loci

[TABLE]

where

[TABLE]

(of course, the sets $\overset{(1)}{\Omega}_{l}$ are continuum ones).

3.3.

Secondly, we use the loci (3.1) for the following values

[TABLE]

see (1.2) where, of course

[TABLE]

Consequently, we define for every admissible and fixed $U,L$ the following three level curves

[TABLE]

as the loci

[TABLE]

3.4.

Finally, construction of the sets $\overset{(1)}{\Omega}_{l},\overset{(2)}{\Omega}_{l}$ together with application of the substitutions (2.3), (3.7) into (1.2) completes the proof of the following.

Theorem 1*.*

For every fixed and admissible $U,L$ (see (1.1), (1.2)) there are continuum sets

[TABLE]

such that for every elements

[TABLE]

we have the following formula (transmutation of (1.2))

[TABLE]

4. Transmutations of types $[\zeta,\cos]$, $[\zeta,(s)^{n_{1}},(s)^{n_{2}},(s)^{n_{3}}]$

4.1.

We use the loci

[TABLE]

for the three values (3.5). Consequently, we define for every admissible and fixed $U,L$ the following level curves

[TABLE]

as the loci

[TABLE]

Now we have the following.

Theorem 2*.*

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

[TABLE]

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

[TABLE]

4.2.

For the functions

[TABLE]

we define the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L,n_{1},n_{2},n_{3}$. Now we have (see (1.2), (3.4), (4.8)) the following.

Theorem 3*.*

For every fixed and admissible $U,L,n_{1},n_{2},n_{3}$ there are the sets

[TABLE]

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

[TABLE]

Remark 12*.*

The formula (4.10) gives an example of infinite set of transmutations of the mother formula (1.2).

5. Transmutations of types $[\zeta,\frac{1}{\Gamma}]$, $[\zeta,J_{p_{1}},J_{p_{2}},J_{p_{3}}]$, $[\zeta,\operatorname{sn},\operatorname{cn},\operatorname{dn}]$

5.1.

For the integral function

[TABLE]

we define the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L$. Now we have (see (1.2), (3.4), (5.3)) the following

Theorem 4*.*

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

[TABLE]

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

[TABLE]

5.2.

For the Bessel’s functions

[TABLE]

we define the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L,p_{1},p_{2},p_{3}$. Now we have (see (1.2), (3.4), (5.8)) the following.

Theorem 5*.*

For every fixed and admissible $U,L,p_{1},p_{2},p_{3}$ there are sets

[TABLE]

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

[TABLE]

Remark 13*.*

Here we have the infinite (enumerable) set of transmutations of transmutations of the mother formula (1.2), comp. Remark 12.

5.3.

For the Jacobi’s elliptic functions (their relief-surfaces with $k=0.8$ see in [1], pp. 202, 203)

[TABLE]

($k_{1},k_{2},k_{3}$ stand for the moduli) we define the level curves

[TABLE]

as the loci

[TABLE]

for every fixed and admissible $U,L,k_{1},k_{2},k_{3}$. Now we have (see (1.2), (3.4), (5.13)) the following.

Theorem 6*.*

For every fixed and admissible $U,L,k_{1},k_{2},k_{3}$ there are sets

[TABLE]

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

[TABLE]

Remark 14*.*

About the potency of the set of transmutations in (5.15) see Remark 2.

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