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Entropy and Its Application to Number Theory

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Seiji Fujino^*

This version is not peer-reviewed

Submitted:

08 January 2024

Posted:

08 January 2024

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Abstract

In this paper, we suggest an expansion of the Planck distribution function derived from the Boltzmann principle. Thereby, we consider expanding Planck's law with a new distribution function, and discuss fine structure constant. Furthermore, using ideas applied to the expansion of the Planck distribution function, we show that Von Koch's inequality can be derived without using the Riemann Hypothesis, that is, the Riemann Hypothesis is true, and that the abc conjecture is not true. Furthermore, we define a generalization of Entropy and discuss that Entropy is relevant to dynamical systems described by logistic function models, such as the growth of bacteria and populations.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

In this paper, we will explain in the following order.

1.1.

First, we introduce the Boltzmann principle and the Planck distribution function to aid understanding. we consider the number of states W that partition particles P into resonators N, and define entropy S using W. Furthermore, the average energy U and a energy element of the resonator

ε

are made to correspond to Entropy S. Thereby, the Planck distribution function and the Planck’s law are derived from Entropy S.

1.2.

Second, we suggest that the expansion of the Planck distribution function which is one of main content on this article. We consider Entropy

S_{π_{f}} (x)

which is the Boltzmann principle divided by function

x / f (x)

, where set the function

f (x)

log (x)

and x be a positive real variable. The function

x / log (x)

is an approximation of the number of prime numbers

π (x)

. Furthermore, the function

R_{α}^{\pm} (x)

is defined and verifies the relationship between

S_{π_{f}} (x)

and

R_{α}^{\pm} (x)

. Thereby, we attempt that the possibility of expanding the Planck distribution function and Planck’s law by using the function

R_{α}^{\pm} (x)

. Besides, the relationship between the constant

α

R_{α}^{\pm} (x)

and fine structure constant will be considered.

1.3.

Third, we consider applying the constant

α

of the function

R_{α}^{\pm} (x)

to number theory. We show that Von Koch’s inequality is derived without using the Riemann hypothesis, that is, it proves the Riemann Hypothesis is correct. Additionally, we show that the abc conjecture is not true.

1.4.

Finally, we will describe some considerations and issues for the future. We generalize Entropy

S_{π_{f}} (x)

to Entropy

S_{D} (x)

, where

D (x)

is a function on x. Using Entropy

S_{D} (x)

, we will discuss that Entropy is related to dynamical systems described by logistic function models, such as bacterial and population growth.

2. The Boltzmann Principle and the Planck Distribution Function

Introduction for Entropy S and the Planck Distribution Function

To make it easier the understanding, we would first let us introduce the Boltzmann principle and the Planck distribution function as follows.

Definition 2.1.

We define symbols using on this article as follows:

\begin{matrix} \begin{matrix} P & : T h e n u m b e r o f p a r t i c l e s, \\ N & : T h e n u m b e r o f r e s o n a t o r s, \\ U & : A v e r a g e e n e r g y p e r a r e s o n a t o r, \\ U_{N} & : T o t a l e n e r g y, \\ ε & : A n e l e m e n t o f e n e r g y, \\ ν & : F r e q u e n c y, \\ T & : T e m p e r a t u r e, \\ k_{B} & : T h e B o l t z m a n n c o n s t a n t, \\ h & : T h e P l a n c k c o n s t a n t, \\ β & : I n v e r s e t e m p e r a t u r e . \end{matrix} \end{matrix}

(1)

□

Using the definitions above, the following equations are satisfied:

\begin{matrix} U_{N} = N U = P ε, t h a t i s, \frac{P}{N} = \frac{U}{ε}, \end{matrix}

(2)

\begin{matrix} β = \frac{1}{k_{B} T}, \end{matrix}

(3)

where the inequality

P > N

is satisfied.

The number of states

W_{N, P}

is defined as the number of particles P is partitioned by the number of partitions

N - 1

and Entropy S (the Boltzmann Principle) is defined as the logarithm of the number of states W, where P and N can be regarded positive integer numbers as follows:

Definition 2.2.

Let the number of particles P and the number of resonators N be positive integers. The number of states and the Boltzmann Principle are defined as follows:

\begin{matrix} W_{N, P} = \frac{(N + P - 1)!}{(N - 1)! P!}, (t h e n u m b e r o f s t a t e s, c o m b i n a t i o n), \end{matrix}

(4)

\begin{matrix} S_{N, P} = k_{B} log W_{N, P}, (B o l t z m a n n P r i n c i p l e), \end{matrix}

(5)

\begin{matrix} S = \frac{S_{N, P}}{N}, (t h e a v e r a g e o f S_{N, P}) . \end{matrix}

(6)

□

Using Stirling’s formula, for sufficiently large natural numbers P and N, the following formulas are satisfied:

\begin{matrix} W_{N, P} = \frac{(N + P - 1)!}{(N - 1)! P!} \approx \frac{{(N + P)}^{N + P}}{N^{N} P^{P}} . \end{matrix}

(7)

Using the Boltzmann principle above, for sufficiently large the number of particles P and resonators N, we can obtain the following equations:

\begin{matrix} \begin{matrix} S_{N, P} & = k_{B} log W_{N, P} \\ = k_{B} \{(N + P) log (N + P) - log N^{N} - log P^{P}\} \\ = k_{B} N \{(1 + \frac{P}{N}) log (1 + \frac{P}{N}) - \frac{P}{N} log \frac{P}{N}\} . \end{matrix} \end{matrix}

(8)

Using the definition the equality (2) and (6) above, the equality (8) is satisfied as follows:

\begin{matrix} \begin{matrix} S & = k_{B} \{(1 + \frac{U}{ε}) log (1 + \frac{U}{ε}) - \frac{U}{ε} log \frac{U}{ε}\} . \end{matrix} \end{matrix}

(9)

Differentiate both sides of the Equation (9) above with respect to average energy per resonator U. Hence, the following equation is satisfied:

\begin{matrix} \begin{matrix} \frac{d S}{d U} & = \frac{k_{B}}{ε} \{log (1 + \frac{U}{ε}) - log \frac{U}{ε}\} . \end{matrix} \end{matrix}

(10)

Furthermore, differentiate both sides of the Equation (10) with respect to average energy per resonator U, the following an equation is satisfied:

\begin{matrix} \begin{matrix} \frac{d^{2} S}{d U^{2}} & = \frac{- k_{B}}{U (ε + U)} . \end{matrix} \end{matrix}

(11)

The rate of change of entropy at energy U ,that is

d S / d U

, is equal to the reciprocal of temperature T. Namely, the following equation is satisfied between Entropy S, average energy per resonator U and temperature T:

\begin{matrix} \begin{matrix} \frac{d S}{d U} & = \frac{1}{T} . \end{matrix} \end{matrix}

(12)

Thus, using the Equations (11) and (12), the following relation is satisfied:

\begin{matrix} \begin{matrix} \frac{d}{d U} (\frac{1}{T}) = \frac{- k_{B}}{U (ε + U)} . \end{matrix} \end{matrix}

(13)

Integrating both sides of the Equation (13) with respect to average energy per resonator U, the following relation is satisfied:

\begin{matrix} \begin{matrix} U & = \frac{ε}{exp (\frac{ε}{k_{B} T}) - 1} . \end{matrix} \end{matrix}

(14)

Here, put

ε

as follows:

\begin{matrix} \begin{matrix} ε & = h ν . \end{matrix} \end{matrix}

(15)

Therefore, the following equations are obtained:

\begin{matrix} \begin{matrix} U & = \frac{h ν}{exp (\frac{h ν}{k_{B} T}) - 1} = \frac{h ν}{exp (h ν β) - 1} . (P l a n c k^{'} s l a w) \end{matrix} \end{matrix}

(16)

The equation above (16) is determined the expression for the average energy of particles in a single mode of frequency

ν

in thermal equilibrium T, that is, named Planck’s law. Besides, we define the distribution function

\bar{n} (ν, β)

as follows:

\begin{matrix} \begin{matrix} \bar{n} (ν, β) & = \frac{1}{exp (h ν β) - 1} . (P l a n c k d i s t r i b u t i o n f u n c t i o n) \end{matrix} \end{matrix}

(17)

This is expressed the mean particle occupation number in thermal equilibrium T . This is named the Planck distribution function on this paper. Moreover, the Equation (17) is transformed as follows:

\begin{matrix} \begin{matrix} \frac{\bar{n} (ν, β)}{\bar{n} (ν, β) + 1} = exp (- h ν β) . (B o l t z m a n n f a c t o r) \end{matrix} \end{matrix}

(18)

The function

exp (- h ν β)

is named the Boltzmann factor. Besides, let

N_{g}

and

N_{e}

be the mean number of atoms in the ground state and in the excited state. The following equation is satisfied:

\begin{matrix} \begin{matrix} \frac{N_{e}}{N_{g}} = exp (- h ν β) . \end{matrix} \end{matrix}

(19)

Note: In this paper, Planck’s radiation law refers to the following expression:

\begin{matrix} U_{p} = \frac{8 π ν^{2}}{c^{3}} \frac{h ν}{exp (h ν β) - 1} . (P l a n c k^{'} s R a d i a t i o n l a w) \end{matrix}

(20)

where the constant c is the speed of light.

3. Expansion of the Planck Distribution Function

We will continue the discussion with reference to ideas in subsection 2. We consider that the number of particles P is replaced to the positive real variable x, and the number of resonator N is replaced to the number of prime number

π (x)

. However the function

π (x)

is not differentiable. Therefore, we consider to partition x by the function

x / log (x)

, that is, divided a positive real variable x by the logarithm

log (x)

. This function

x / log (x)

is an approximation of

π (x)

. Namely, the number of resonator N is replaced to the number of prime number

x / log (x)

Thus, we show that the function

R_{α}^{\pm} (x)

and expansion of the Planck distribution function are derived as follows.

3.1. Entropy $S_{π_{f}}$ that Partitioned by Approximation of $π (x)$

First, we start with the definition of Entropy

S_{π_{f}} (x)

on a positive real variable x.

Definition 3.1.

Let

x > 1

be a positive real variable, and

f (x)

be a positive real valued function on x.

\begin{matrix} \begin{matrix} π (x) & = \sum_{\begin{matrix} p \leq x \\ p : p r i m e \end{matrix}} 1, \end{matrix} \end{matrix}

(21)

\begin{matrix} π_{f} (x) = \frac{x}{f (x)}, \end{matrix}

(22)

\begin{matrix} Q_{f} (x) = \frac{x}{π_{f} (x)} . \end{matrix}

(23)

The function

π (x)

is expressed that the number of prime numbers less than or equal to x. By the definition above, it is satisfied that

Q_{f} (x) = f (x)

. □

We define the number of states

W_{π_{f}, x}

. Entropy

S_{π_{f}, x}

under

W_{π_{f}, x}

is defined by the number of states

W_{π_{f}, x}

. Moreover, Entropy

S_{π_{f}} (x)

under

π_{f} (x)

is defined to devided by Entropy

S_{π_{f}, x}

π_{f} (x)

as follows:

Definition 3.2.

We define that the number of state

W_{π_{f}, x}

, Entropy

S_{π_{f, x}}

and Entropy

S_{π_{f}} (x)

. Let

x > 1

be a positive real variable.

\begin{matrix} W_{π_{f}, x} = \frac{{(π_{f} (x) + x)}^{π_{f} (x) + x}}{π_{f} {(x)}^{π_{f} (x)} x^{x}}, \end{matrix}

(24)

\begin{matrix} S_{π_{f}, x} = log W_{π_{f}, x}, \end{matrix}

(25)

\begin{matrix} S_{π_{f}} (x) = \frac{S_{π_{f}, x}}{π_{f} (x)} . \end{matrix}

(26)

□

Note: Since the definition of Combination below Formula (27) cannot define real values well, therefore, we adopted the definition of Formula (24) using Stirling’s approximation.

\begin{matrix} \frac{(π_{f} (x) + x - 1)!}{(π_{f} (x) - 1)! x!} . \end{matrix}

(27)

In the following discussion, unless otherwise specified, let the function

f (x)

set to

log (x)

. Namely, the following is satisfied:

\begin{matrix} f (x) = log (x) . \end{matrix}

(28)

Therefore, using definitions above and the prime number theorem (Refer to Narkiewicz [1]), the following conditions are satisfied:

\begin{matrix} Q_{f} (x) = Q_{log} (x) = \frac{x}{π_{log} (x)} = log (x) \sim \frac{x}{π (x)} . \end{matrix}

(29)

By the Definition 3.2, for

x > 1

, the following equations are satisfied:

\begin{matrix} \begin{matrix} S_{π_{f}, x} & = (π_{f} (x) + x) log (π_{f} (x) + x) - π_{f} (x) log (π_{f} (x)) - x log (x) \\ = π_{f} (x) ((1 + \frac{x}{π_{f} (x)}) log (1 + \frac{x}{π_{f} (x)}) - \frac{x}{π_{f} (x)} log (\frac{x}{π_{f} (x)})), \end{matrix} \end{matrix}

(30)

\begin{matrix} \begin{matrix} S_{π_{f}} (x) & = (1 + \frac{x}{π_{f} (x)}) log (1 + \frac{x}{π_{f} (x)}) - \frac{x}{π_{f} (x)} log (\frac{x}{π_{f} (x)}) . \end{matrix} \end{matrix}

(31)

Using the function

Q_{f} (x)

above, the function

S_{π_{f}} (x)

is expressed as follows:

\begin{matrix} \begin{matrix} S_{π_{f}} (x) & = (1 + Q_{f} (x)) log (1 + Q_{f} (x)) - Q_{f} (x) log Q_{f} (x) . \end{matrix} \end{matrix}

(32)

Differentiating Entropy

S_{π_{f}} (x)

, the following formulas are satisfied.

\begin{matrix} \begin{matrix} S_{π_{f}}^{'} (x) & = {(\frac{x}{π_{f} (x)})}^{'} log (1 + \frac{x}{π_{f} (x)}) + {(\frac{x}{π_{f} (x)})}^{'} \\ - ({(\frac{x}{π_{f} (x)})}^{'} log (\frac{x}{π_{f} (x)}) + {(\frac{x}{π_{f} (x)})}^{'}) \\ = {(\frac{x}{π_{f} (x)})}^{'} (log (1 + \frac{x}{π_{f} (x)}) - log (\frac{x}{π_{f} (x)})) . \end{matrix} \end{matrix}

(33)

Furthermore, differentiating

S_{π_{f}}^{'} (x)

as follows:

\begin{matrix} \begin{matrix} S_{π_{f}}^{″} (x) & = {(\frac{x}{π_{f} (x)})}^{″} (log (1 + \frac{x}{π_{f} (x)}) - log (\frac{x}{π_{f} (x)})) \\ + {(\frac{x}{π_{f} (x)})}^{'} {(log (1 + \frac{x}{π_{f} (x)}) - log (\frac{x}{π_{f} (x)}))}^{'} . \end{matrix} \end{matrix}

(34)

Therefore, the equations above is expressed by using

Q_{f} (x)

as follows:

\begin{matrix} S_{π_{f}}^{'} (x) = Q_{f}^{'} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)), \end{matrix}

(35)

\begin{matrix} \begin{matrix} S_{π_{f}}^{″} (x) = Q_{f}^{″} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) \\ + Q_{f}^{'} (x) Q_{f}^{'} (x) (\frac{- 1}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix} \end{matrix}

(36)

Repeating differential of the part of

Q_{f}^{''} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x))

on (36), the following conditions are satisfied:

\begin{matrix} \begin{matrix} {(Q_{f}^{''} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)))}^{'} \\ = Q_{f}^{(3)} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) \\ + Q_{f}^{(2)} (x) Q_{f}^{'} (x) (\frac{- 1}{Q_{f} (x) (1 + Q_{f} (x))}), \end{matrix} \end{matrix}

(37)

\begin{matrix} \begin{matrix} {(Q_{f}^{(n)} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)))}^{'} \\ = Q_{f}^{(n + 1)} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) \\ + Q_{f}^{(n)} (x) Q^{'} {(x)}_{f} (\frac{- 1}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix} \end{matrix}

(38)

Therefore, for all

x > 1

, the following conditions are satisfied:

\begin{matrix} \begin{matrix} C a s e (1), & n > 1 i s e v e n n u m b e r : \\ Q_{f}^{(n + 1)} (x) = \frac{{(- 1)}^{n} (n)!}{x^{n + 1}} > \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} = Q_{f}^{(n)} (x), \end{matrix} \end{matrix}

(39)

\begin{matrix} \begin{matrix} C a s e (2), & n > 1 i s o d d n u m b e r : \\ Q_{f}^{(n + 1)} (x) = \frac{{(- 1)}^{n} (n)!}{x^{n + 1}} < \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} = Q_{f}^{(n)} (x) . \end{matrix} \end{matrix}

(40)

Furthermore, for all

x > 1

, the following are satisfied:

\begin{matrix} Q_{f}^{(n)} (x) & > & Q_{f}^{(2)} (x), \end{matrix}

(41)

\begin{matrix} |Q_{f}^{(n + 1)} (x)| & > & |Q_{f}^{(n)} (x)|, \end{matrix}

(42)

where n is a positive integer.

Next, we define some functions

k_{f} (x)

R_{m}^{+} (x)

and

R_{m}^{-} (x)

as follows:

Definition 3.3.

The definition of the function

k_{f} (x)

Let

x > 1

be a positive real variable, and

f (x)

be a real valued function. The function

k_{f} (x)

is defined such that satisfies the following equation:

\begin{matrix} k_{f} (x) = S_{π_{f}}^{''} (x) (\frac{- Q_{f} (x) (1 + Q_{f} (x))}{Q_{f}^{'} (x)}) . \end{matrix}

(43)

Therefore, the following equation is satisfied:

\begin{matrix} S_{π_{f}}^{''} (x) = k_{f} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix}

(44)

□

The function

k_{f} (x)

is named the Boltzmann

k_{f}

function.

Definition 3.4.

The function

R_{m}^{+} (x)

and

R_{m}^{-} (x)

are defined as follows:

\begin{matrix} R_{m}^{+} (x) = \sum_{n = 1}^{m} \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} . \end{matrix}

(45)

Therefore, the following equations are satisfied:

\begin{matrix} R_{m}^{+} (x) = \sum_{n = 1}^{m} \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} = \sum_{n = 1}^{m} {(log (x))}^{(n)} . \end{matrix}

(46)

Same as discussion, the following inequality are satisfied:

\begin{matrix} R_{m}^{-} (x) = \sum_{n = 1}^{m} |\frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}}| . \end{matrix}

(47)

Therefore, the following equations are satisfied:

\begin{matrix} R_{m}^{-} (x) = \sum_{n = 1}^{m} |\frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}}| = \sum_{n = 1}^{m} {|(log (x))}^{(n)} | . \end{matrix}

(48)

□

The function

R_{m}^{+} (x)

is named an m-th absolute lower approximation of

k_{f} (x)

or simply

R_{m}^{+}

-function. Similarly, the function

R_{m}^{-} (x)

is named an m-th lower approximation of

k_{f} (x)

or simply

R_{m}^{-}

-function. Using the defintion above, the following inequality is satisfied:

\begin{matrix} R_{m}^{-} (x) \geq R_{m}^{+} (x), \end{matrix}

(49)

where the function

{(log (x))}^{(n)}

represents the n-th differentiation of

log (x)

Note: To be distinguish from the function

{(log (x))}^{(n)}

, the n-th power

log (x)

is represented the function

{(log (x))}^{n}

and

{log}^{n} (x)

Using the Equation (44), for all

x > 1

, the following conditions are satisfied:

\begin{matrix} k_{f} (x) = - S_{π_{f}}^{''} (x) \frac{Q_{f} (x) (1 + Q_{f} (x))}{Q_{f}^{'} (x)} \leq \frac{1}{x} (2 + log (x)) . \end{matrix}

(50)

where the function

Q_{f} (x)

log (x)

by the definition. Because, by the Equation (36),

\begin{matrix} \begin{matrix} S_{π_{f}}^{″} (x) = Q_{f}^{″} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) \\ + Q_{f}^{'} (x) Q_{f}^{'} (x) (\frac{- 1}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix} \end{matrix}

(51)

Therefore, for

x > 1

, the following are satisfied:

\begin{matrix} \begin{matrix} k_{f} (x) & = \frac{1}{x^{2}} log (1 + \frac{1}{log (x)}) x log (x) (1 + log (x)) + \frac{1}{x} \\ = \frac{1}{x} log {(1 + \frac{1}{log (x)})}^{log (x)} (1 + log (x)) + \frac{1}{x} \\ \leq \frac{1}{x} log (e) (1 + log (x)) + \frac{1}{x} ∵) lim_{x \to \infty} {(1 + \frac{1}{log (x)})}^{log (x)} \to e \\ = \frac{1}{x} (1 + log (x)) + \frac{1}{x} \\ = \frac{1}{x} (2 + log (x)) . \end{matrix} \end{matrix}

(52)

Furthermore, there exist a positive integer

m \geq 1

such that the following conditions are satisfied:

\begin{matrix} \begin{matrix} S_{π_{f}}^{″} (x) & = k_{f} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}) \\ \geq (Q_{f}^{'} (x) + Q_{f}^{''} (x) + \dots + Q_{f}^{(m)} (x)) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} \\ \geq R_{m}^{+} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix} \end{matrix}

(53)

Using the same discussion above, there is a positive integer

m \geq 1

such that the following conditions are satisfied:

\begin{matrix} \begin{matrix} S_{π_{f}}^{″} (x) & = k_{f} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}) \\ \geq (| Q_{f}^{'} (x) | + | Q_{f}^{''} (x) | + \dots + | Q_{f}^{(m)} (x) |) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} \\ \geq R_{m}^{-} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}) . \end{matrix} \end{matrix}

(54)

The first order differentiation of Entropy

S_{π_{f}} (x)

is always positive values, that is

S_{π_{f}}^{'} (x) > 0

. Moreover, the second order differentiation of Entropy

S_{π_{f}} (x)

has always negative values, so that

S_{π_{f}}^{''} (x) < 0

. Therefore, Entropy

S_{π_{f}} (x)

has no inflection points.

3.2. Derivation of the Functions $R_{α}^{\pm}$

Next, the function

R_{α}^{+} (x)

and

R_{α}^{-} (x)

are derived as follows:

Definition 3.5.

R_{α}^{+} (x)

R_{α}^{-} (x)

and

R_{α}^{\pm} (x)

Let the constant

α > 0

be a positive real number. For all positive real variable

x > 1

, the function

R_{α}^{+} (x)

and

R_{α}^{-} (x)

are defined as follows:

\begin{matrix} R_{α}^{\pm} (x) = \frac{\sqrt{2 π} α}{e x \pm 1} . \end{matrix}

(55)

Therefore, the following conditions are satisfied:

\begin{matrix} x R_{α}^{\pm} (x) = \frac{\sqrt{2 π} α x}{e x \pm 1}, \end{matrix}

(56)

\begin{matrix} \frac{1}{x R_{α}^{\pm} (x)} = \frac{e}{\sqrt{2 π} α} (1 \pm \frac{1}{e x}) . \end{matrix}

(57)

□

This function

R_{α}^{\pm} (x)

is a lower approximation of

k_{f} (x)

This function

R_{α}^{\pm} (x)

is named

R_{α}^{\pm}

lower approximation by

k_{f} (x)

and

α

or simply

R_{α}^{\pm}

-function. The relations of functions

R_{α}^{\pm} (x)

are satisfied as follows:

Lemma 3.6.

The relation

R_{m}^{\pm} (x) \geq R_{α}^{\pm} (x)

Let the constant

α > 0

be a positive real number. There exist an integer

m \geq 1

such that for sufficiently large real variable

x > 1

, the following inequality is satisfied:

\begin{matrix} R_{m}^{\pm} (x) \geq R_{α}^{\pm} (x) = \frac{\sqrt{2 π} α}{e x \pm 1} \end{matrix}

(58)

where a positive real number α are satisfied as follows:

\begin{matrix} x \geq \frac{\pm}{\sqrt{2 π} α - e}, \end{matrix}

(59)

that is, satisfied as follows:

\begin{matrix} \frac{e}{\sqrt{2 π}} (1 \pm \frac{1}{e x}) \geq α . \end{matrix}

(60)

Proof.

The proof of Lemma 3.6 are described the following the Section 6.1. □

Consequently, for all real variable

x > 1

, a real valued function

f (x)

and a positive integer

m > 1

, the following inequalities are satisfied:

\begin{matrix} S_{π_{f}}^{''} (x) \geq R_{m}^{\pm} (x) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} \geq R_{α}^{\pm} (x) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}, \end{matrix}

(61)

Namely, the following inequality is satisfied:

\begin{matrix} S_{π_{f}}^{''} (x) \geq R_{α}^{\pm} (x) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} . \end{matrix}

(62)

The second order differentiation of Entropy

S_{π_{f}}^{''} (x)

is suppressed from the bottom side by Formula (62). Besides, the function

R_{α}^{\pm} (x)

is suppressed from the upper side by formula as follows:

Lemma 3.7.

The upper upper approximation of

R_{m}^{\pm} (x)

For sufficiently large real variable

x > 1

and a positive integer

m > 1

, the following inequalities are satisfied:

\begin{matrix} \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e \pm 1} \geq R_{m}^{\pm} (x) \geq R_{α}^{\pm} (x) = \frac{\sqrt{2 π} α}{e x \pm 1} . \end{matrix}

(63)

Proof.

The proof of Lemma 3.7 are described the following the Section 6.2. □

Note: The function

R_{α} \pm

and the constant

α

are derived by dividing the entropy space x by the approximation

x / log (x)

π (x)

and by applying Starling’s formula to the series obtained by the n-th differentiation of

Q_{f} (x)

. These are reasons that is defined as the set of irrational numbers

\sqrt{2}

, e, and

π

On the next subsection, we discuss the meaning of inequalities in inequality (62).

3.3. The Expanded Planck Distribution Function $n^{\pm} (x, α)$

Next, we examine to define the expanded Planck distribution functions

n^{\pm} (x, α)

by using

R_{α}^{\pm} (x)

. Integrating the inequality (61) by a variable x, the following formulas are satisfied:

\begin{matrix} \int S_{π_{f}}^{''} (x) d x \geq \int R_{α}^{\pm} (x) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} d x . \end{matrix}

(64)

Beside, the following are satisfied:

\begin{matrix} S_{π_{f}}^{'} (x) = \int S_{π_{f}}^{''} (x) d x + C, \end{matrix}

(65)

\begin{matrix} Q_{f}^{'} (x) \geq R_{α}^{\pm} (x) . \end{matrix}

(66)

Therefore, the following formulas are satisfied:

\begin{matrix} \begin{matrix} S_{π_{f}}^{'} (x) & = Q_{f}^{'} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) + C \\ \geq R_{α}^{\pm} (x) (log (1 + Q_{f} (x)) - log Q_{f} (x)) + C \\ = R_{α}^{\pm} (x) log (1 + \frac{1}{Q_{f} (x)}) + C . \end{matrix} \end{matrix}

(67)

where the constant C is a positive real number.

Here, for all sufficiently large

Q_{f} (x) > 0

, the following equation is satisfied:

\begin{matrix} log (1 + \frac{1}{Q_{f} (x)}) = 0 . \end{matrix}

(68)

Hence, the first order differentiation

S_{π_{f}}^{'} (x)

is satisfied as follows:

\begin{matrix} S_{π_{f}}^{'} (x) = Q_{f}^{'} (x) log (1 + \frac{1}{Q_{f} (x)}) = 0 . \end{matrix}

(69)

Thus, the constant C is satisfied as follows:

\begin{matrix} C = 0 . \end{matrix}

(70)

Therefore, the inequality (67) is satisfied as follows:

\begin{matrix} S_{π_{f}}^{'} (x) \geq R_{α}^{\pm} (x) log (1 + \frac{1}{Q_{f} (x)}) . \end{matrix}

(71)

For positive real variable

x > 1

, the function

S_{π_{f}}^{'} (x)

are satisfied as follows:

\begin{matrix} \frac{1}{x} \geq S_{π_{f}}^{'} (x) = \frac{1}{x} log (1 + \frac{1}{log (x)}) . \end{matrix}

(72)

According to inequalities (71) and (72), the following is satisfied:

\begin{matrix} \frac{1}{x R_{α}^{\pm} (x)} \geq log (1 + \frac{1}{Q_{f} (x)}) . \end{matrix}

(73)

Therefore, by (73) the following inequality is derived:

\begin{matrix} Q_{f} (x) \geq \frac{1}{exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1} . \end{matrix}

(74)

Focusing on the equality part of the inequality (74), we define new distribution function

n^{\pm} (x, α)

as follows:

Definition 3.8.

The expanded Planck distribution function

n^{\pm} (x, α)

The expanded distribution functions

n^{\pm} (x, α)

of the Planck distribution function

\bar{n} (ν, β)

are defined as follows:

\begin{matrix} n^{\pm} (x, α) = \frac{1}{exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1}, \end{matrix}

(75)

where

α > 0

. □

This distribute function

n^{\pm} (x, α)

is named the expanded Planck distribution function. The definition above (75) is transformed as follows:

\begin{matrix} \frac{n^{\pm} (x, α)}{n^{\pm} (x, α) + 1} = exp (\frac{- 1}{x R_{α}^{\pm} (x)}), (α > 0) . \end{matrix}

(76)

Thus, this function

n^{\pm} (x, α)

can be regarded as one of the distribution functions. Therefore, the expanded Planck distribution functions

n^{\pm} (x, α)

are regards as the approximate density of

x / π (x)

until the number x. Furthermore, this function

n^{\pm} (x, α)

is seems to be expanded the Planck distribution function

\bar{n} (ν, β)

. According to imitate the Boltzmann factor, the following function

\begin{matrix} exp (\frac{- 1}{x R_{α}^{\pm} (x)}) \end{matrix}

(77)

is named the expanded Boltzmann factor or

R_{α}^{\pm}

-factors. We will consider the further relationship in the next subsection.

3.4. Correspondence the Planck Distribution Function $\bar{n} (ν, β)$ and the Expanded Planck Distribution Function $n^{\pm} (x, α)$

We examine to correspond the Planck distribution function

\bar{n} (ν, β)

and the expanded Planck distribution function

n^{\pm} (x, α)

as follows:

\begin{matrix} \bar{n} (ν, β) = \frac{1}{exp (h ν β) - 1}, (P l a n c k d i s t r i b u t i o n f u n c t i o n) \end{matrix}

(78)

where h: the Planck constant ,

ν

: Frequency,

β

: Inverse temperature.

Here, we consider to correspond the internal parameter of the Boltzmann factor

exp (- h ν β)

\begin{matrix} h ν β \end{matrix}

(79)

and the internal function of

R_{α}^{\pm}

-factor

exp (\frac{- 1}{x R_{α}^{\pm} (x)})

\begin{matrix} \frac{1}{x R_{α}^{\pm} (x)} (= \frac{e}{\sqrt{2 π} α} (1 \pm \frac{1}{e x})) . \end{matrix}

(80)

Namely, we suppose the correspondence as follows:

\begin{matrix} h ν β = \frac{e}{\sqrt{2 π} α} (1 \pm \frac{1}{e x}) . \end{matrix}

(81)

Furthermore, we can consider by separating the correspondence between and the variable parts and the constant parts as follows:

\begin{matrix} ν β = (1 \pm \frac{1}{e x}), (v a r i a b l e p a r t s) \end{matrix}

(82)

\begin{matrix} h = \frac{e}{\sqrt{2 π} α} . (c o n s t a n t p a r t s) \end{matrix}

(83)

The relationship diagram between

S_{N, P}

and

S_{π_{f}, x}

is shown below:

\begin{matrix} \begin{matrix} W_{N, P} = \frac{(N + P - 1)!}{(N - 1)! P!}, & S_{N, P} = k_{B} log W_{N, P}, \\ exp (- h ν β) & \overset{log}{⟶} & - h ν β \\ \overset{N : = π_{f} (x)}{P : = x} 🡓 🡑 \overset{α : = \frac{e}{\sqrt{2 π} h}}{x : = \frac{\pm 1}{e (ν β - 1)}} & \overset{N : = π_{f} (x)}{P : = x} 🡓 🡑 \overset{α : = \frac{e}{\sqrt{2 π} h}}{x : = \frac{\pm 1}{e (ν β - 1)}} \\ W_{π_{f}, x} = \frac{{(π_{f} (x) + x)}^{π_{f} (x) + x}}{π_{f} {(x)}^{π_{f} (x)} x^{x}}, & S_{π_{f}, x} = log W_{π_{f}, x}, \\ exp (\frac{- 1}{x R_{α}^{\pm} (x)}) & \overset{log}{⟶} & \frac{- 1}{x R_{α}^{\pm} (x)} \end{matrix} \end{matrix}

(84)

Corresponding the above, the expanded Planck distribution function

n^{\pm} (x, α)

becomes to expand the Planck distribution function

\bar{n} (ν, β)

. Namely, the following conditions are satisfied:

Suggestion 3.9.

The expansion of the Planck distribution

\bar{n} (ν, β)

Let the constant

α > 0

be a positive real number. For all real variable

x > 1

the following equation is satisfied:

\begin{matrix} \begin{matrix} \bar{n} (ν, β) & = n^{\pm} (x, α), \end{matrix} \end{matrix}

(85)

where

\begin{matrix} \begin{matrix} x & = \frac{\pm 1}{e (ν β - 1)}, t h a t i s, ν β = (1 \pm \frac{1}{e x}), \\ α & = \frac{e}{\sqrt{2 π} h}, t h a t i s, h = \frac{e}{\sqrt{2 π} α} . \end{matrix} \end{matrix}

(86)

Namely, the distribution function

n^{\pm} (x, α)

can be regarded as representing an expansion of the Planck distribution function

\bar{n} (ν, β)

. □

For sufficiently large

x > 1

, the correspondence of Equation (82) is satisfied as follows:

\begin{matrix} ν β = lim_{x \to \infty} (1 \pm \frac{1}{e x}) = 1 . \end{matrix}

(87)

Moreover, according to the method to divide each S and

S_{π_{f}} (x)

, we remember that the following corresponds:

(1): The number of particles P is replaced to the positive real variable x.
(2): The number of resonators N is replaced to the approximate of $π (x)$ , that is, $\frac{x}{log (x)}$ .

We remember that the following conditions are satisfied:

\begin{matrix} \frac{U}{ε} = \frac{P}{N} \end{matrix}

(88)

\begin{matrix} Q_{f} (x) = \frac{x}{\frac{x}{log (x)}} = \frac{1}{\frac{1}{log (x)}} (\sim \frac{x}{π (x)}) . \end{matrix}

(89)

Namely, we suppose the correspondence as follows:

\begin{matrix} \begin{matrix} U & ⟷ x ⟷ 1, \\ ε & ⟷ \frac{x}{log (x)} ⟷ \frac{1}{log (x)} . \end{matrix} \end{matrix}

(90)

Thereby, we define the following function

U_{x, α}^{\pm}

corresponding to Planck’s law U.

Definition 3.10.

The function

U_{x, α}^{\pm}

as the expansion of Planck’s law U.

Let the constant

α > 0

be a positive real number. For sufficiently large real variable

x > 1

, the real valued function

U_{x, α}^{\pm}

is defined as follows:

\begin{matrix} \begin{matrix} U_{x, α}^{\pm} & = n^{\pm} (x, α) \frac{1}{log (x)} = \frac{1}{log (x) (exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1)} \end{matrix} \end{matrix}

(91)

□

Using suggestion 3.9 and Definition 3.10, the following suggestion is obtained:

Suggestion 3.11.

The expansion of Planck’s law U.

Let

h > 0

ν > 0

and

β > 0

be real numbers. Each values h, ν and β means the Planck constant, frequency and inverse temperature.

There exists a real variable

x > 1

and a constant

α > 0

such that the following equality is satisfied:

\begin{matrix} \begin{matrix} U = U_{x, α}^{\pm}, \end{matrix} \end{matrix}

(92)

where the following conditions are satisfied:

\begin{matrix} \begin{matrix} h ν = \frac{1}{log (x)}, ν = \frac{1}{h log (x)}, h = \frac{e}{\sqrt{2 π} α}, \\ β = h log (x) (1 \pm \frac{1}{e x}) . \end{matrix} \end{matrix}

(93)

Proof.

The above proposal is satisfied by setting as follows. For any

ν > 0

\begin{matrix} x = exp (\frac{1}{h ν}) ≒ exp (\frac{1}{6.626 \times 10^{- 34} ν}) > 1, \end{matrix}

(94)

where the Planck constant

h ≒ 6.626 \times 10^{- 34} m^{2} k g / s

. □

Namely, the real valued function

U_{x, α}^{\pm}

can be regarded as representing the expansion of Planck’s law U. The function

U_{x, α}^{\pm}

is named the expanded Planck’s law.

As a result of suggestion 3.11 above, energy

h ν

is put

1 / log (x)

. Namely, Planck’s law U is seems to has an spectrum such that an reciprocal of logarithm. It is possible that the discrete values of an element of energy are related to the distribution of prime numbers. The relationship diagram between Planck’s law U and the expanded Planck’s law

U^{\pm} (x, α)

is shown below:

\begin{matrix} \begin{matrix} S_{N, P} = k_{B} log W_{N, P}, & U = \frac{h ν}{exp (\frac{h ν}{k_{B} T}) - 1}, \\ B o l t z m a n n P r i n c i p l e & \overset{\frac{U}{ε}, S^{''}}{⟶} & P l a n c k^{'} s l a w \\ \overset{N : = π_{f} (x)}{P : = x} 🡓 🡑 \overset{α : = \frac{e}{\sqrt{2 π} h}}{x : = \frac{\pm 1}{e (ν β - 1)}} & \overset{N : = π_{f} (x)}{P : = x} 🡓 🡑 \overset{α : = \frac{e}{\sqrt{2 π} h}}{h ν : = \frac{1}{log (x)}} \\ S_{π_{f}, x} = log W_{π_{f}, x}, & U^{\pm} (x, α) = \frac{1}{log (x)} \frac{1}{exp (\frac{1}{R_{α}^{\pm} (x)}) - 1}, \\ S_{π_{f}, x} - E n t r o p y & \overset{\frac{1}{log (x)}, S_{π}^{''} (x)}{⟶} & t h e e x p a n d e d P l a n c k^{'} s l a w \end{matrix} \end{matrix}

(95)

3.5. Fine Structure Constants

The above section, the expanded Planck distribution

n^{\pm} (x, α)

and the Planck distribution

\bar{n} (ν, β)

was related by a constant

α

. In this section, we discuss that the constant

α = e / \sqrt{2 π} h

is thought like the fine structure constant.

Let a constant

α

set as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π} h} . \end{matrix}

(96)

For all sufficiently large

x > 1

, the following inequalities are satisfied:

\begin{matrix} \frac{e}{\sqrt{2 π}} (2 + log (x)) \geq α, \end{matrix}

(97)

that is, take

x > 1

as follows:

\begin{matrix} x \geq exp (\frac{1}{h} - 2) > 1 . \end{matrix}

(98)

According to the Prime numbers theorem, the following relation is satisfied:

\begin{matrix} π (x) \sim \frac{x}{log (x)} \sim \frac{x}{log (x) + 2} . \end{matrix}

(99)

Thus, the constant

α

is satisfied such that

\begin{matrix} \frac{e}{\sqrt{2 π}} \frac{x}{π (x)} \geq α, (α > 0) . \end{matrix}

(100)

Hence, the positive real number

α

can be regard as fine structure constant by

\sqrt{2}

π, e

and

π (x)

. Furthermore, the following inequality is satisfied:

Suggestion 3.12.

The relation between the Planck constant h and

π (x)

There exists a positive real number

α > 0

such that for sufficiently large

x > 1

, the following formulas are satisfied:

\begin{matrix} \begin{matrix} h = \frac{e}{\sqrt{2 π} α} \geq \frac{1}{2 + log (x)} \sim \frac{π (x)}{x} . \end{matrix} \end{matrix}

(101)

Namely, for sufficiently large

x > 1

, the Planck constant h is bigger than the ratio of a positive real number x and

π (x)

. □

Therefore, it is considered that the constant

α

is related between the Planck distribution function

\begin{matrix} \bar{n} (ν, β) = \frac{1}{exp (h ν β) - 1}, \end{matrix}

(102)

and the expanded Planck distribution function

\begin{matrix} n^{\pm} (x, α) = \frac{1}{exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1} . \end{matrix}

(103)

Specially, suppose the constant

h_{α}

and

α_{h}

are decided by

α

, e and

π

as follows:

\begin{matrix} h_{α} = \frac{e}{\sqrt{2 π} α}, \end{matrix}

(104)

\begin{matrix} α_{h} = \frac{e}{\sqrt{2 π} h} . \end{matrix}

(105)

Therefore, by suggestion 3.9, Modern physics may be a special case that satisfy the following conditions:

\begin{matrix} α_{h} = α = \frac{e}{\sqrt{2 π} h}, \end{matrix}

(106)

\begin{matrix} h_{α} = h = \frac{e}{\sqrt{2 π} α} . \end{matrix}

(107)

In other words, the following suggestion is stated:

Suggestion 3.13.

Let the constant

α > 0

be a positive real number. The constant

h_{α}

can be selected as follows:

\begin{matrix} h_{α} = \frac{e}{\sqrt{2 π} α}, \end{matrix}

(108)

where the following inequality is satisfied:

\begin{matrix} \frac{e}{\sqrt{2 π}} \frac{x}{π (x)} \geq α, (α > 0) . \end{matrix}

(109)

Namely, if the condition is satisfied as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π} h}, \end{matrix}

(110)

then the constant

h_{α}

becomes the Planck constant h. □

Note:

Let me mention here for your attention as follows: The fine structure constant is a physical constant

α

and is originally expressed as following (111) using the Planck constant. In this paper, we describe it as original the fine structure constant

α_{p}

to distinguish it from the constant

α

. Besides, describe it as the elementary charge

e_{p}

to distinguish it from Napier’s number e. original the fine structure constant

α_{p}

is follows:

\begin{matrix} α_{p} = \frac{e_{p}^{2}}{2 h ε_{0} c} = \frac{μ_{0} e_{p}^{2} c}{2 h}, . \end{matrix}

(111)

where

\begin{matrix} \begin{matrix} h & : t h e P l a n c k c o n s t a n t, \\ ε_{0} & : t h e e l e c t r i c c o n s t a n t, \\ μ_{0} & : t h e m a g n e t i c c o n s t a n t, \\ e_{p} & : t h e e l e m e n t a r y c h a r g e, \\ c & : t h e s p e e d o f l i g h t . \end{matrix} \end{matrix}

(112)

Therefore, the relation original the fine structure constant

α_{p}

and the constant

α

is satisfied as follows:

\begin{matrix} \begin{matrix} \frac{α_{p}}{α} & = \sqrt{\frac{π}{2 e^{2}}} \frac{e_{p}^{2}}{ε_{0} c} = \sqrt{\frac{π}{2 e^{2}}} e_{p}^{2} μ_{0} c . \end{matrix} \end{matrix}

(113)

The suggestion 3.13 is seems obvious. However, on the following section, we show that some examples such that the constant

α \neq \frac{e}{\sqrt{2 π} h}

as follows:

\begin{matrix} α = \frac{1}{\sqrt{2 π}}, α = \frac{2}{\sqrt{2 π}}, α = \frac{e}{\sqrt{2 π}}, α = \frac{e}{\sqrt{2 π} log (\frac{ϵ + 2}{ϵ + 1})} . \end{matrix}

(114)

4. Application the Function $R_{α}^{\pm}$ to Number Theory

4.1. Examples Using the Function $R_{α}^{+}$ for Deriving Von Koch’s Inequality

We derive Von Koch’s inequality using the above the constant

α

and the function

R_{α}^{+} (x)

. (Refer to Fujino [2] for the proof)

Note: In this paper, we consider an element of energy

ε

and an arbitrary real number

ϵ

separately.

Theorem 4.1.

Inequalities for evaluating the number of prime numbers (1).

Let

α > 0

be a positive real number (constant). There exist a positive real number

C > 1

such that for all sufficiently large real variable

x \geq 2 (\in R)

, the following conditions are satisfied:

\begin{matrix} | π (x) - li (x) | \leq C {(\frac{\sqrt{2 π} α}{48})}^{\frac{1}{4}} exp (\frac{e}{\sqrt{2 π} α}) x^{\frac{1}{\sqrt{2 π} α}} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}), \end{matrix}

(115)

where the positive real number

α > 0

are satisfied as follows:

\begin{matrix} 1 \leq \frac{1}{\sqrt{2 π} α} exp (\frac{e}{\sqrt{2 π} α}), \end{matrix}

(116)

\begin{matrix} \frac{1}{\sqrt{2 π}} \leq α \leq C \frac{e}{\sqrt{2 π}}, \end{matrix}

(117)

\begin{matrix} exp (\frac{e}{\sqrt{2 π} α}) = lim_{x \to \infty} exp (\frac{1}{x R_{α}^{+} (x)}) \leq exp (\frac{e}{\sqrt{2 π} α} (1 + \frac{1}{e x})) . \end{matrix}

(118)

Proof.

See [3] for the proof. □

Corollary 4.2.

Inequalities for evaluating the number of prime numbers (2).

There exist a positive real number

C > 1

such that for all

ϵ > 0 (\in R)

and for all sufficiently large

x \geq 2 (\in R)

, the following conditions is satisfied:

\begin{matrix} | π (x) - li (x) | \leq C {(\frac{1}{48})}^{\frac{1}{4}} exp (e) x {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) . \end{matrix}

(119)

Proof.

Using Theorem 4.1, put a positive real number

α > 0

as follows:

\begin{matrix} α = \frac{1}{\sqrt{2 π}} . \end{matrix}

(120)

Therefore, the inequality (119) is satisfied.

□

The result of Corollary 4.1 is similar to the following result:(Refer to Narkiewicz [1])

\begin{matrix} (\exists C > 0) | π (x) - li (x) | \leq O (x exp (- C \sqrt{l o g (x)})) . \end{matrix}

(121)

Comparing inequalities (119) and (121), the following condition is satisfied:

\begin{matrix} O (x {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)})) \leq O (x exp (- C \sqrt{l o g (x)})) . \end{matrix}

(122)

Namely, the asymptotic formula of (119) gives better than that of (121).

Therefore, let

α = 2 / \sqrt{2 π}

(h_{α} = e / 2)

. the Theorem 4.1 is satisfied as follows:

Corollary 4.3.

Inequalities for evaluating the number of prime numbers (3).

There exist a positive real number

C > 1

such that for all

ϵ > 0 (\in R)

and for all sufficiently large

x \geq 2 (\in R)

, the following condition is satisfied:

\begin{matrix} | π (x) - li (x) | \leq C {(\frac{1}{24})}^{\frac{1}{4}} exp (\frac{e}{2}) x^{\frac{1}{2}} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) . \end{matrix}

(123)

Proof.

Using Theorem (4.1), the following conditions is satisfied:

\begin{matrix} 1 \leq \frac{1}{\sqrt{2 π} α} exp (\frac{e}{\sqrt{2 π} α}) . \end{matrix}

(124)

Put a positive real number

α > 0

as follows:

\begin{matrix} α = \frac{2}{\sqrt{2 π}} (\geq \frac{1}{\sqrt{2 π}}) . \end{matrix}

(125)

Hence, the following inequalities is satisfied:

\begin{matrix} \begin{matrix} 1 & \leq \frac{1}{\sqrt{2 π} α} exp (\frac{e}{\sqrt{2 π} α}) \\ = \frac{1}{\sqrt{2 π} \frac{2}{\sqrt{2 π}}} exp (\frac{e}{\sqrt{2 π} \frac{2}{\sqrt{2 π}}}) (∵ α = \frac{2}{\sqrt{2 π}}) \\ = \frac{1}{2} exp (\frac{e}{2}) (= 1.946424 \dots) . \end{matrix} \end{matrix}

(126)

Thus, the positive real number

α > 0

is satisfied conditions of (124) and (125). Therefore, there exist a positive real number

C > 1

such that for all sufficiently large

x \geq 2

the following condition is satisfied:

\begin{matrix} | π (x) - li (x) | \leq C {(\frac{1}{24})}^{\frac{1}{4}} exp (\frac{e}{2}) x^{\frac{1}{2}} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) . \end{matrix}

(127)

□

Corollary 4.4.

Von Koch’s inequality.

\begin{matrix} (\exists C > 1) (\forall ϵ > 0) (\forall x > > 2) | π (x) - li (x) | \leq C x^{\frac{1}{2}} log (x), \end{matrix}

(128)

where

C, ϵ

and x are real numbers. Namely,

\begin{matrix} | π (x) - li (x) | \leq O (x^{\frac{1}{2}} log (x)) . \end{matrix}

(129)

Proof.

Fix

ϵ > 0

. For all sufficient large

x \geq 2

, the following conditions are satisfied:

\begin{matrix} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) < log (x) < x^{ϵ} . \end{matrix}

(130)

Therefore, there exist a positive real number

C > 0

such that for all sufficiently large

x \geq 2

, the following inequalities are satisfied:

\begin{matrix} \begin{matrix} | π (x) - li (x) | \\ \leq C {(\frac{1}{24})}^{\frac{1}{4}} exp (\frac{e}{2}) x^{\frac{1}{2}} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) \\ \leq C x^{\frac{1}{2}} log (x) . \end{matrix} \end{matrix}

(131)

□

As well known, Corollary 4.4 is equivalent to the Riemann Hypothesis. (Refer to Narkiewicz [1]) Therefore, the Riemann Hypothesis is true.

Furthermore, let

α = e / \sqrt{2 π}

(h_{α} = 1)

. The following inequality is satisfied:

Corollary 4.5.

The reduction of the upper limit of the inequality that evaluate the number of prime number.

\begin{matrix} (\exists C > 1) (\forall ϵ > 0) (\forall x > > 2) | π (x) - li (x) | \leq C x^{\frac{1}{e}} log (x), \end{matrix}

(132)

where

C, ϵ

and x are real numbers

(\in R)

. Namely,

\begin{matrix} | π (x) - li (x) | \leq O (x^{\frac{1}{e}} log (x)) . \end{matrix}

(133)

Proof.

Using Theorem 4.1 , put a positive real number

α > 0 (\in R)

as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π}} (\geq \frac{e}{\sqrt{2 π}}) . \end{matrix}

(134)

Thus, the following conditions are satisfied:

\begin{matrix} \begin{matrix} 1 & \leq \frac{1}{\sqrt{2 π} α} exp (\frac{e}{\sqrt{2 π} α}) \\ = \frac{1}{\sqrt{2 π} \frac{e}{\sqrt{2 π}}} exp (\frac{e}{\sqrt{2 π} \frac{e}{\sqrt{2 π}}}) (∵ α = \frac{e}{\sqrt{2 π}}) \\ = \frac{1}{e} exp (1) (= 1) . \end{matrix} \end{matrix}

(135)

Therefore the following condition is satisfied:

\begin{matrix} \begin{matrix} | π (x) - li (x) | \\ \leq C {(\frac{e}{48})}^{\frac{1}{4}} exp (1) x^{\frac{1}{e}} {(\frac{1}{log (x)})}^{\frac{3}{4}} exp (\frac{1}{log (x)}) \\ \leq C x^{\frac{1}{e}} log (x) . \end{matrix} \end{matrix}

(136)

□

We have a question that the upper limit of the real number d that satisfies the formula

| π (x) - li (x) | \leq O (x^{1 / d} log (x))

is Napier’s constant e correct or not. Namely, the following problem can be considered.

Problem 4.6.

The upper limit of the inequality that evaluate the number of prime number.

\begin{matrix} e = sup {d > 1 ∣ (\exists C > 1) (\forall ϵ > 0) (\forall x > > 2) | π (x) - li (x) | \leq C x^{\frac{1}{d}} log (x)}, \end{matrix}

(137)

where

C, ϵ

and x are real numbers

(\in R)

. Namely,

\begin{matrix} e = sup {d > 1 ∣ | π (x) - li (x) | \leq O (x^{\frac{1}{d}} log (x))} . \end{matrix}

(138)

□

We expect problem 4.6 to be correct. In future, we attempt to solve this problem.

4.2. Example Using the Function $R_{α}^{-}$ for the abc Conjecture

We derive the negation of the weak abc conjecture and the strong abc conjecture using the constant

α

and the function

R_{α}^{-} (x)

. Namely, using the constant

α

and the function

R_{α}^{-} (x)

, the following theorems are satisfied: (Refer to Fujino [4] for the proof)

Theorem 4.7.

Let

α > 0

be a positive real number. For all real number

ϵ > 0

and the constant

K_{ϵ} \geq 1

, there exists countable infinite triples

(a, b, c)

of coprime positive integers with

a + b = c

such that the following inequality is satisfied:

\begin{matrix} K_{ϵ} rad (a b c) < c^{exp (\frac{e}{\sqrt{2 π} α}) - 1} \end{matrix}

(139)

where

α > 0

is satisfied as follows:

\begin{matrix} exp (\frac{e}{\sqrt{2 π} α}) = lim_{x \to \infty} exp (\frac{1}{x R_{α}^{-} (x)}) \geq exp (\frac{e}{\sqrt{2 π} α} (1 - \frac{1}{e x})) \end{matrix}

(140)

□

Set the constant

α

as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π} log (\frac{ϵ + 2}{ϵ + 1})} . \end{matrix}

(141)

Therefore, the following is satisfied:

\begin{matrix} h_{α} = log (\frac{ϵ + 2}{ϵ + 1}) . \end{matrix}

(142)

Therefore, the negation of the weak abc conjecture is satisfied as follows:

Theorem 4.8.

The negation of the weak abc conjecture.

For all real number

ϵ > 0

and the constant

{\bar{K}}_{ϵ} \geq 1

, there exists countable infinite triples

(a, b, c)

of coprime positive integers with

a + b = c

such that the following inequality is satisfied:

\begin{matrix} {\bar{K}}_{ϵ} rad {(a b c)}^{1 + ϵ} < c . \end{matrix}

(143)

Namely, There exist a counter-example in the weak abc conjecture. Therefore, the weak abc conjecture is not true. □

Furthermore, let

ϵ = 1

and

{\bar{K}}_{ϵ} = 1

. The negation of the strong abc conjecture is satisfied as follows:

Theorem 4.9.

The negation of the strong abc conjecture.

There exists countable infinite triples

(a, b, c)

of coprime positive integers with

a + b = c

such that the following inequality is satisfied:

\begin{matrix} rad {(a b c)}^{2} < c . \end{matrix}

(144)

Namely, the strong abc conjecture is not true. □

The function

R_{α}^{\pm} (x)

was used the above discussions of Von Koch’s inequality and the abc conjecture. As mentioned above, Entropy and Number theory are thought to be closely related.

4.3. Conclusion and Application to Number Theory

The above discussion, we attempted to apply the Boltzmann principle and the Planck distribution function to prime number theory. We considered dividing natural number x by an approximation of

π (x)

, that is,

x / log (x)

. Thereby, we obtained that the function

R_{α}^{\pm} (x)

. Furthermore, using the function

R_{α}^{\pm} (x)

, we derived and define new distribution function

n^{\pm} (x, α)

As mentioned above,modern physics is considered to be the special condition that the real number

α

be satisfied as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π} h}, that is, h_{α} = h, \end{matrix}

(145)

Furthermore, using the expanded Planck distribution function

R_{α}^{\pm}

, we evaluated Von Koch’s inequality that equivalent to the Riemann Hypothesis and the abc conjecture. Namely, we considered the system different from modern physics such that Von Koch’s inequality is satisfied as follows:

\begin{matrix} α = \frac{2}{\sqrt{2 π}}, that is, h_{α} = \frac{e}{2}, \end{matrix}

(146)

and, the abc conjecture is satisfied as follows:

\begin{matrix} α = \frac{e}{\sqrt{2 π} log (\frac{ϵ + 2}{ϵ + 1})}, that is, h_{α} = log (\frac{ϵ + 2}{ϵ + 1}) . \end{matrix}

(147)

We believe that there exist closely relationship between entropy and number theory. It may be possible there exists the system that satisfies new constant

h_{α}

or non-constant

k_{f} (x)

, which is different from the constants of modern physics such as Planck’s constant and Boltzmann’s constant.

5. Generalization of Entropy and Application to Dynamical Systems

In this section, we focus on the generalization of entropy and the second order derivative of entropy. In addition, we will consider that entropy is related to dynamical systems described by logistic function models, such as bacterial and population growth.

5.1. About $S_{π_{f}}^{'} (x)$ and $S_{π_{f}}^{''} (x)$

We consider the generalization of

S_{π_{f}}^{'} (x)

and

S_{π_{f}}^{''} (x)

(Refer to Nicolis, Prigogine [5,6]). Let

x > 1

be a real variable. These functions

S_{π_{f}} (x)

S_{π_{f}}^{'} (x)

and

S_{π_{f}}^{''} (x)

above are regarded as follows:

\begin{matrix} \begin{matrix} S_{π_{f}} (x) : E n t r o p y p a r t i t i o n e d b y π_{f} (x), \\ S_{π_{f}}^{'} (x) : E n t r o p y v e l o c i t y S_{π_{f}} (x), (E n t r o p y g e n e r a t i o n) \\ S_{π_{f}}^{″} (x) : E n t r o p y a c c e l e r a t i o n S_{π_{f}} (x) . \end{matrix} \end{matrix}

(148)

where f,

Q_{f}

and

π_{f}

are satisfied as follows:

\begin{matrix} \begin{matrix} f (x) = log (x), \\ Q_{f} (x) = log (x), \\ π_{f} (x) = \frac{x}{f (x)} = \frac{x}{log (x)} . \end{matrix} \end{matrix}

(149)

The first order derivative of function

S_{π_{f}} (x)

, that is,

S_{π_{f}}^{'} (x)

and the second order derivative of function

S_{π_{f}} (x)

, that is,

S_{π_{f}}^{''} (x)

can be describe as follows:

\begin{matrix} \begin{matrix} S_{π_{f}}^{'} (x) = Q_{f}^{'} (x) log (1 + \frac{1}{Q_{f} (x)}), \\ S_{π_{f}}^{″} (x) = k_{f} (x) (\frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))}), \end{matrix} \end{matrix}

(150)

where the function

k_{f} (x)

is regarded as a function decided by the function

π_{f} (x)

and

Q_{f} (x)

(Refer to Definitiosn (36) and (43)). The function

Q_{f} (x)

can be regarded as the position divided a real variable x by

Q_{f} (x)

. The first order derivative of

Q_{f} (x)

, that is,

Q_{f}^{'} (x)

is the slope of

Q_{f} (x)

or the change

Q_{f}^{'} (x)

of the position

f (x)

. Entropy velocity can be regraded as Entropy generation (Refer to Nicolis, Prigogine [5,6]). We attempt to consider the generalization of Entropy below subsection.

5.2. Generalization of Entropy Acceleration $S_{D}^{''} (x)$

We generalize the equation above (64) as follows. Let

x > 1

be real variable and

ξ > 0

be a constant. The function

D (x)

be a positive real valued function such that

D (x) \leq x

. The function

D (x)

can be thought of as a division of x. Therefore, the above

S_{D} (x)

S_{D}^{'} (x)

and

S_{D}^{''} (x)

are regarded and defined as follows:

\begin{matrix} S_{D} (x) = (1 + \frac{Q_{D} (x)}{ξ}) log (1 + \frac{Q_{D} (x)}{ξ}) - \frac{Q_{D} (x)}{ξ} log \frac{Q_{D} (x)}{ξ}, \end{matrix}

(151)

\begin{matrix} S_{D}^{'} (x) = \frac{Q_{D}^{'} (x)}{ξ} log (1 + \frac{ξ}{Q_{D} (x)}), \end{matrix}

(152)

\begin{matrix} S_{D}^{''} (x) = k_{D} (x) (\frac{- Q_{D}^{'} (x)}{Q_{D} (x) (ξ + Q_{D} (x))}), \end{matrix}

(153)

where the relation between the functions

D (x)

and

Q_{D} (x)

are satisfied as follows:

\begin{matrix} D (x) = \frac{ξ x}{Q_{D} (x)} . \end{matrix}

(154)

The function

k_{D} (x)

can be regard as a function decided by x,

ξ

and

D (x)

. The function

Q_{D} (x)

can be regard as a position divided a real value

ξ x

Q_{D} (x)

. The first order derivative of

Q_{D} (x)

, that is,

Q_{D}^{'} (x)

can be regard as the change of the position by x and

ξ

. Each functions above are real valued functions.

5.3. Relationship between $S_{D}^{''} (x)$ and $S_{π_{f}}^{''} (x)$

Regarding as the correspondence between the generalized Equation (153) of

S_{D}^{''} (x)

and the Equation (64) of

S_{π_{f}}^{''} (x)

, we can put it as follows:

\begin{matrix} \begin{matrix} S_{D}^{″} (x) & : = S_{π_{f}}^{″} (x) (< 0), \\ Q_{D} (x) & : = Q_{f} (x), \\ Q_{D}^{'} (x) & : = Q_{f}^{'} (x) . \\ k_{D} (x) & : = R_{α}^{\pm} (x), \\ ξ & : = 1 . \end{matrix} \end{matrix}

(155)

Therefore, we can obtain the following equation:

\begin{matrix} S_{π_{f}}^{''} (x) d x = R_{α}^{\pm} (x) \frac{- Q_{f}^{'} (x)}{Q_{f} (x) (1 + Q_{f} (x))} d x . \end{matrix}

(156)

This Equation (156), that is, (64), can be regarded as an example of applying

Q_{f}

to the Equation (153). In the subsections below, we examine some examples can be regarded as the accelerations of

S_{D} (x)

, that is, (153).

5.4. Logistic Function of Bacterial, Population Growth and Generalize Entropy

We examine the relationship between generalized the function

S_{D}^{''} (x)

of equation(153) and the logistic function, such as the model of bacterial and population growth. (Refer to R. May [7], N. Bacaër [8], M,Yamaguchi [9])

Let r and K be positive real constants. For positive real number

t > 0

, let

N (t)

be a positive real valued function. We put as follows:

\begin{matrix} \begin{matrix} S_{D}^{″} (x) & : = - r \\ Q_{D} (x) & : = - \frac{N (t)}{K}, \\ Q_{D}^{'} (x) & : = - \frac{1}{K} \frac{d N (t)}{d t}, \\ k_{D} (x) & : = 1, \\ x & : = t, \\ ξ & : = 1, \end{matrix} \end{matrix}

(157)

where parameters r, K, t and the function

N (x)

mean as follows:

\begin{matrix} \begin{matrix} r & : t h e g r o w t h r a t e, \\ N (t) & : b a c t e r i a l o r p o p u l a t i o n g r o w t h \\ K & : c a r r y i n g c a p a c i t y, \\ t & : t i m e o r s t e p . \end{matrix} \end{matrix}

(158)

Hence, the Equation (153) becomes the equation of the dynamical system as follows:

\begin{matrix} \int - r d t \geq \int \frac{- (\frac{- 1}{K})}{- \frac{N (t)}{K} (1 - \frac{N (t)}{K})} \frac{d N (t)}{d t} d t . \end{matrix}

(159)

Transforming the formula above as follows:

\begin{matrix} \int r d t \geq \int \frac{K}{N (t) (K - N (t))} d N (t) . \end{matrix}

(160)

Therefore, the following equation is obtained:

\begin{matrix} \frac{d N (t)}{d t} = r N (t) \frac{(K - N (t))}{K} . \end{matrix}

(161)

By the way, the solution of logistic Equation (161) above are satisfied as follows:

\begin{matrix} N (t) = \frac{K}{1 + exp (- (r t + C))}, \end{matrix}

(162)

where C is a constant.

The Equation (161) above is the logistic function of dynamics. In other words, the Equation (161) derived from the Equation (153) can be regarded as an application of the dynamical system. Namely, we can consider the following possibilities:

For

x > 1

, a constant

ξ > 0

and a function

D (x)

, the Equation (153), that is,

\begin{matrix} S_{D}^{''} (x) = k_{D} (x) (\frac{- Q_{D}^{'} (x)}{Q_{D} (x) (ξ + Q_{D} (x))}), \end{matrix}

is regarded as a generalized expression of the equation(161), that is,

\begin{matrix} \frac{d N (t)}{d t} = r N (t) \frac{(K - N (t))}{K}, \end{matrix}

where the function

D (x)

needs to be chosen appropriately such as above example.

Therefore, entropy and dynamical systems are considered to be closely related.

5.5. Complex Dynamic Systems.

We introduce the relationship between the logistic Function (161) above and complex dynamic systems as follows.

(Refer to R.M May [7], N.Bacaër [8] and M.Yamaguchi [9])

Definition 5.1.

Let the constant a be a real number that satisfies

0 < a \leq 4

and

x_{n}

be real numbers between [0,1] where

n \geq 0

is positive integers.

(x_{n} \in [0, 1], n \geq 0)

The discrete dynamical system is defined as follows:

\begin{matrix} x_{n + 1} = a x_{n} (1 - x_{n}) . \end{matrix}

(163)

□

As is known, the Equation (161) can be expressed like the Equation (163) as follows. Namely, the Equation (161) is rewrite as a difference equation as follows:

\begin{matrix} \frac{N (t + Δ t) - N (t))}{Δ t} = r N (t) \frac{(K - N (t))}{K} . \end{matrix}

(164)

Put as follows:

\begin{matrix} N_{n} = N (n Δ t) . \end{matrix}

(165)

Using (165), the Equation 164 is expressed in the difference equation of

N_{n}

as follows:

\begin{matrix} N_{n + 1} = ((1 + r Δ t) - \frac{r Δ t}{K} N_{n}) N_{n} . \end{matrix}

(166)

Therefore, the following is satisfied:

\begin{matrix} \frac{r Δ t}{K (1 + r Δ t)} N_{n + 1} = (1 + r Δ t) (1 - \frac{r Δ t}{K (1 + r Δ t)} N_{n}) \frac{r Δ t}{K (1 + r Δ t)} N_{n} . \end{matrix}

(167)

Furthermore, we consider the following corresponds:

\begin{matrix} \begin{matrix} x_{n} : = \frac{r Δ t}{K (1 + r Δ t)} N_{n}, \\ a : = (1 + r Δ t) . \end{matrix} \end{matrix}

(168)

Using correspondence of the Equation (168), we can transform the Equation (167) to the following equation:

\begin{matrix} x_{n + 1} = a x_{n} (1 - x_{n}) . \end{matrix}

(169)

This Equation (169) above is same as the Equation (163) in the Definition 5.1. According to the results of Li, Yorke and R.M.May et al. (Refer to [7,10]), depending on how the number

0 < a \leq 4

is taken, a complicated orbit, that is, chaos appear. The discussion so far can be summarized as follows:

(1): Logistic function (161) is derived from Entropy acceleration $S_{D}^{''} (x)$ of (153),
(2): Complex dynamic systems (169) is derived from Logistic function (161),
(3): Chaos is derived from Complex dynamic systems (169).

Logistic function and Complex dynamic systems are special cases of Entropy acceleration. Complex dynamic systems produces periodic order depending on how its conditions are taken. On the other hand, Complex dynamic systems create chaos depending on how their conditions are set. In other words, Entropy not only increases, but also has the potential to create chaos and order.

5.6. Conclusions

Entropy was related to statistical quantum mechanics by Boltzmann and Planck. By partitioning the average energy of a resonator by its energy component (or partitioning the number of particles by the number of resonators), the Planck distribution and Planck’s law are derived and these are expressed the actual physical phenomenon that describes the spectrum of black-body radiation.

We defined Entropy

S_{π_{f}} (x)

by partitioning the Boltzmann principle by an approximate of

π (x)

(that is,

x / log (x)

). Thereby, we discussed that the expanded Planck distribution function, the expanded Planck’ law and fine structure constants. Although no examples of physical phenomena expressing the extended Planck distribution and the extended Planck’s law have been found, is it possible that there exists a constant that satisfies a constant other than Planck’s constant?

In addition, we applied to number theory by using the constant

α

and the function

R_{α}^{\pm} (x)

. Namely, by using the constant

α = 2 / \sqrt{2 π}

R_{α}^{+}

-factor

exp (\frac{1}{R_{α}^{+} (x)})

, we derive Van Koch’s inequality that is equivalent to the Riemann hypothesis. In addition, by using the constant

α = e / \sqrt{2 π} log (\frac{ϵ + 2}{ϵ + 1})

R_{α}^{-}

-factor and

exp (\frac{1}{R_{α}^{-} (x)})

, it could be applied to be derived the negation of the abc conjecture.

These examples of number theory are not actual physical phenomena, but the constants

α

and

R_{α}^{\pm}

-factor that satisfy extended Planck distribution and extended Planck’s law exists in number theory created by the human mind.

If Number theory created by the human mind allow as a part of physical phenomena, then it may be considered to express a physical phenomenon that is “a kind of radiations that comes out from inside humans” . In any case, We hope to discover examples of physical phenomena related to each fine structure constant.

Furthermore, Entropy was related with complex systems by Prigogine and Nicolis, et al. We attempted the generalization of Entropy

S_{D} (x)

, and discussed that Entropy acceleration

S_{D}^{''} (x)

(the second order differentiation of Entropy

S_{D} (x)

) can be related to dynamical systems described by logistic function models such as bacterial and population growth.

We are sure there are other potentially useful applications. Therefore, we hope that it is worth continuing to explore these researches, and that this paper will serve as a bridge to further research and that Entropy will be further studied and many things will develop in the future.

Increasing Entropy (the second law) does not mean becoming disordered. On the contrary, the second law has the potential to cause movement and order in phenomena. (Refer to Fujino [11])

The following figure shows the relationship between several fields and entropy.

6. The proof of Lemma 3.6 and Lemma 3.7

6.1. The proof of Lemma 3.6

Proof.

First, we show the proof of

R_{m}^{-} (x)

Let n

\geq 1

a positive integer. For all sufficiently large positive real variable

x > 0

, the following conditions are satisfied:

\begin{matrix} \begin{matrix} | {(Q_{f} (x))}^{(n)} | & = | \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} | \\ = \frac{(n - 1)!}{x^{n}} \\ \geq \frac{\sqrt{2 π} {(n - 1)}^{(n - 1 + \frac{1}{2})} e^{- (n - 1)}}{x^{n}} \\ (∵ {Stirling}^{'} s formula : n! \geq \sqrt{2 π} e^{- n} n^{n + \frac{1}{2}}) \\ = (\frac{\sqrt{2 π} {(n - 1)}^{n - (\frac{1}{2})}}{e^{(n - 1)} x^{n}}) \\ = \sqrt{2 π} {(n - 1)}^{n - (\frac{1}{2})} \frac{1}{e^{(n - 1)} x^{n}} (* 2) . \end{matrix} \end{matrix}

(170)

Therefore, dividing the end of the Formula (170), that is (*2), by the number

n^{n - (\frac{1}{2})}

, for sufficiently large

x > 1

, the following condition are satisfied:

Case 1)

x > n

: Because

x \geq x - (\frac{1}{2})

, therefore

\begin{matrix} \begin{matrix} (* 2) & \geq \sqrt{2 π} {(\frac{n - 1}{n})}^{n - (\frac{1}{2})} \frac{1}{e^{(n - 1)} x^{n}} \\ \geq \sqrt{2 π} {(\frac{x - 1}{x})}^{x - (\frac{1}{2})} \frac{1}{e^{(n - 1)} x^{n}} (∵ x > n) \\ \geq \sqrt{2 π} {(\frac{x - 1}{x})}^{x} \frac{1}{e^{(n - 1)} x^{n}} \\ (∵ x \geq x - (\frac{1}{2}) and {(\frac{x - 1}{x})}^{x} \geq lim_{x \to \infty} {(\frac{x - 1}{x})}^{x} = e^{- 1}) \\ \geq \sqrt{2 π} e^{- 1} \frac{1}{e^{(n - 1)} x^{n}} . \end{matrix} \end{matrix}

(171)

Case 2)

n \geq x

\begin{matrix} \begin{matrix} (* 2) & \geq \sqrt{2 π} {(\frac{n - 1}{n})}^{n - (\frac{1}{2})} \frac{1}{e^{(n - 1)} x^{n}} \\ \geq \sqrt{2 π} {(\frac{n - 1}{n})}^{n} \frac{1}{e^{(n - 1)} x^{n}} \\ (∵ n \geq n - (\frac{1}{2}) and {(\frac{n - 1}{n})}^{n} \geq lim_{n \to \infty} {(\frac{n - 1}{n})}^{n} = e^{- 1}) \\ \geq \sqrt{2 π} e^{- 1} \frac{1}{e^{(n - 1)} x^{n}} . \end{matrix} \end{matrix}

(172)

Therefore, using Case 1) and Case 2) above, for all sufficiently large

x > 0

, the following inequality is satisfied:

\begin{matrix} | {(Q_{f} (x))}^{(n)} | \geq \sqrt{2 π} e^{- 1} \frac{1}{e^{(n - 1)} x^{n}} . \end{matrix}

(173)

Therefore, the following conditions are satisfied:

\begin{matrix} \begin{matrix} R_{m}^{-} (x) & \geq lim_{N \to \infty} \sum_{n = 1}^{N} |{(Q_{f} (x))}^{(n)}| \\ \geq lim_{N \to \infty} \sqrt{2 π} e^{- 1} \sum_{n = 1}^{N} |\frac{{(- 1)}^{n - 1}}{e^{(n - 1)} x^{n}}| \\ \geq lim_{N \to \infty} \sqrt{2 π} e^{- 1} \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{e^{(n - 1)} x^{n}} . \end{matrix} \end{matrix}

(174)

Put the function

A (x)

as follows:

\begin{matrix} A (x) : = \sum_{n = 1}^{N} |\frac{{(- 1)}^{n - 1}}{e^{(n - 1)} x^{n}}| . \end{matrix}

(175)

Expand the series

A (x)

above as follows:

\begin{matrix} A (x) & = & \frac{1}{e^{0} x^{1}} + \frac{1}{e^{1} x^{2}} + \frac{1}{e^{2} x^{3}} + \dots + |\frac{{(- 1)}^{N - 1}}{e^{N - 1} x^{N}}|, \end{matrix}

(176)

\begin{matrix} \frac{1}{e x} A (x) & = & \frac{1}{e^{1} x^{2}} + \frac{1}{e^{2} x^{3}} + \frac{1}{e^{3} x^{4}} + \dots + |\frac{{(- 1)}^{N - 1}}{e^{N} x^{N + 1}}| . \end{matrix}

(177)

Subtract (177) from (176). Hence, the following conditions are satisfied:

\begin{matrix} (1 - \frac{1}{e x}) A (x) = lim_{N \to \infty} (\frac{1}{x} - \frac{{(- 1)}^{N - 1}}{e^{N} x^{N + 1}}) = \frac{1}{x} . \end{matrix}

(178)

Therefore, it is satisfied as follows:

\begin{matrix} lim_{N \to \infty} A (x) = \frac{1}{x} (\frac{1}{1 - \frac{1}{e x}}) = \frac{e}{e x - 1} . \end{matrix}

(179)

Therefore, for integer

m > 1

, the function

R_{m}^{-}

is approximated as follows:

\begin{matrix} R_{m}^{-} (x) \geq \sqrt{2 π} e^{- 1} lim_{N \to \infty} A (x) = \frac{\sqrt{2 π}}{e x - 1} . \end{matrix}

(180)

Here, the following conditions are satisfied:

\begin{matrix} R_{m}^{-} (x) > \frac{1}{x} > \frac{\sqrt{2 π} α}{e x - 1} = R_{α}^{-} (x) . \end{matrix}

(181)

Put

α > 0

such that as follows:

\begin{matrix} x \geq \frac{- 1}{\sqrt{2 π} α - e} . \end{matrix}

(182)

Consequently, the following conditions are satisfied:

\begin{matrix} R_{m}^{-} (x) \geq \sqrt{2 π} e^{- 1} α lim_{N \to \infty} A (x) = \frac{\sqrt{2 π} α}{e x - 1} = R_{α}^{-} (x), \end{matrix}

(183)

\begin{matrix} x \geq \frac{- 1}{\sqrt{2 π} α - e}, that is \frac{e}{\sqrt{2 π}} (1 - \frac{1}{e x}) \geq α . \end{matrix}

(184)

For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} \frac{1}{x} (2 + log (x)) \geq k_{f} (x) \geq R_{α}^{-} (x) = \frac{\sqrt{2 π} α}{e x - 1}, \end{matrix}

(185)

\begin{matrix} \frac{e}{\sqrt{2 π}} (2 + log (x)) \geq \frac{e}{\sqrt{2 π}} (1 - \frac{1}{e x}) \geq α > 0, \end{matrix}

(186)

where

α > 0

is a real number. (The end of the proof

R_{m}^{-} (x)

on Lemma 3.6)

By the same as methods, the following conditions are satisfied:

\begin{matrix} \begin{matrix} R_{m}^{+} (x) & \geq lim_{N \to \infty} \sum_{n = 1}^{N} {(Q_{f} (x))}^{(n)} \\ \geq lim_{N \to \infty} \sqrt{2 π} e^{- 1} \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{e^{(n - 1)} x^{n}} . \end{matrix} \end{matrix}

(187)

Because, for all sufficiently large

x > 0

, the following inequality is satisfied:

\begin{matrix} i f n i s e v e n : {(Q_{f} (x))}^{(n)} \geq \sqrt{2 π} e^{- 1} \frac{{(- 1)}^{n}}{e^{(n - 1)} x^{n}}, \end{matrix}

(188)

\begin{matrix} i f n i s o d d : {(Q_{f} (x))}^{(n)} \leq \sqrt{2 π} e^{- 1} \frac{{(- 1)}^{n}}{e^{(n - 1)} x^{n}} . \end{matrix}

(189)

Therefore the condition (187) is satisfied. Put the function

B (x)

as follows:

\begin{matrix} B (x) : = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{e^{(n - 1)} x^{n}} . \end{matrix}

(190)

Expand the series

B (x)

above as follows:

\begin{matrix} B (x) & = & \frac{1}{e^{0} x^{1}} - \frac{1}{e^{1} x^{2}} + \frac{1}{e^{2} x^{3}} - \dots + \frac{{(- 1)}^{N - 1}}{e^{N - 1} x^{N}}, \end{matrix}

(191)

\begin{matrix} \frac{1}{e x} B (x) & = & \frac{1}{e^{1} x^{2}} - \frac{1}{e^{2} x^{3}} + \frac{1}{e^{3} x^{4}} - \dots + \frac{{(- 1)}^{N - 1}}{e^{N} x^{N + 1}} . \end{matrix}

(192)

Add the Equation (191) and (192). Hence, it is satisfied as follows:

\begin{matrix} (1 + \frac{1}{e x}) B (x) = lim_{N \to \infty} (\frac{1}{x} + \frac{{(- 1)}^{N - 1}}{e^{N} x^{N + 1}}) = \frac{1}{x} . \end{matrix}

(193)

Therefore, it is satisfied as follows:

\begin{matrix} lim_{N \to \infty} B (x) = \frac{1}{x} (\frac{1}{1 + \frac{1}{e x}}) = \frac{e}{e x + 1} . \end{matrix}

(194)

Here, the following conditions are satisfied:

\begin{matrix} R_{m}^{+} (x) > \frac{1}{x} > \frac{\sqrt{2 π} α}{e x + 1} = R_{α}^{+} (x) . \end{matrix}

(195)

Put

α > 0

such that as follows:

\begin{matrix} x \geq \frac{1}{\sqrt{2 π} α - e} . \end{matrix}

(196)

Consequently, the following conditions are satisfied:

\begin{matrix} R_{m}^{+} (x) \geq \sqrt{2 π} e^{- 1} α lim_{N \to \infty} B (x) = \frac{\sqrt{2 π} α}{e x + 1} = R_{α}^{+} (x), \end{matrix}

(197)

\begin{matrix} x \geq \frac{1}{\sqrt{2 π} α - e}, that is \frac{e}{\sqrt{2 π}} (1 + \frac{1}{e x}) \geq α . \end{matrix}

(198)

For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} \frac{1}{x} (2 + log (x)) \geq k_{f} (x) \geq R_{α}^{+} (x) = \frac{\sqrt{2 π} α}{e x + 1}, \end{matrix}

(199)

\begin{matrix} \frac{e}{\sqrt{2 π}} (2 + log (x)) \geq \frac{e}{\sqrt{2 π}} (1 + \frac{1}{e x}) \geq α > 0, \end{matrix}

(200)

where

α > 0

is a real number. (The end of the proof

R_{m}^{+} (x)

on Lemma 3.6) □

6.2. The Proof of Lemma 3.7

Proof.

First, we show the proof of

R_{m}^{+} (x)

Let

n \geq 1

and

x > 0

is sufficiently large.

\begin{matrix} \begin{matrix} | {(Q_{f} (x))}^{(n)} | & = | \frac{{(- 1)}^{n - 1} (n - 1)!}{x^{n}} | \\ = \frac{(n - 1)!}{x^{n}} \\ \leq \frac{e {(n - 1)}^{(n - 1 + \frac{1}{2})} e^{- (n - 1)}}{x^{n}} \\ (∵ {Stirling}^{'} s formula : e e^{- n} n^{n + \frac{1}{2}} \geq n! \geq \sqrt{2 π} e^{- n} n^{n + \frac{1}{2}}) \\ = (\frac{e {(n - 1)}^{n - (1 / 2)}}{e^{(n - 1)} x^{n}}) \\ = e {(n - 1)}^{n - (1 / 2)} \frac{1}{e^{(n - 1)} x^{n}} \\ = e {(\frac{n - 1}{x})}^{n} \frac{{(n - 1)}^{- 1 / 2}}{e^{(n - 1)}} \\ = e {(\frac{x - 1}{x})}^{x} \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} (x ≫ n) \\ \leq e {(\frac{x - 1}{x})}^{x} \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} \\ \leq e e^{- 1} \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} (∵ lim_{x \to \infty} {(\frac{x - 1}{x})}^{x} = \frac{1}{e}) \\ \leq \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix} \end{matrix}

(201)

Therefore, the following inequality are satisfied:

\begin{matrix} | {(Q_{f} (x))}^{(n)} | \leq \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix}

(202)

Furthermore, the inequality

\begin{matrix} | {(Q_{f} (x))}^{(n)} | > | {(Q_{f} (x))}^{(n + 1)} | \end{matrix}

(203)

is satisfied. Besides, the following relation are satisfied:

\begin{matrix} i f n i s e v e n : 0 < {(Q_{f} (x))}^{(n)} \leq \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}}, \end{matrix}

(204)

\begin{matrix} i f n i s o d d : 0 > {(Q_{f} (x))}^{(n)} \geq \frac{- 1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix}

(205)

The discussion above, the following are satisfied:

\begin{matrix} \begin{matrix} R_{m}^{+} (x) & = lim_{N \to \infty} \sum_{n = 1}^{N} {(Q_{f} (x))}^{(n)} \leq lim_{N \to \infty} \sum_{n = 1}^{N} \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix} \end{matrix}

(206)

Put the function

B_{2} (x)

as follows:

\begin{matrix} B_{2} (x) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix}

(207)

Hence, we can describe as follows:

\begin{matrix} B_{2} (x) = \frac{1}{e^{0} {(x - 1)}^{1 / 2}} - \frac{1}{e^{1} {(x - 1)}^{1 / 2}} + \dots - \frac{{(- 1)}^{N - 1}}{e^{N - 1} {(x - 1)}^{1 / 2}}, \end{matrix}

(208)

\begin{matrix} \frac{1}{e} B_{2} (x) = \frac{1}{e^{1} {(x - 1)}^{1 / 2}} - \frac{1}{e^{2} {(x - 1)}^{1 / 2}} + \dots - \frac{{(- 1)}^{N - 1}}{e^{N} {(x - 1)}^{1 / 2}} \end{matrix}

(209)

Add the equivalent (208) and (209), the following equivalent are satisfied:

\begin{matrix} \begin{matrix} (1 + \frac{1}{e}) B_{2} (x) & = lim_{N \to \infty} (\frac{1}{{(x - 1)}^{1 / 2}} + \frac{{(- 1)}^{N}}{e^{N} {(x - 1)}^{1 / 2}}) = \frac{1}{{(x - 1)}^{1 / 2}} . \end{matrix} \end{matrix}

(210)

Therefore, it is satisfied as follows:

\begin{matrix} \begin{matrix} lim_{N \to \infty} B_{2} (x) & = \frac{1}{{(x - 1)}^{1 / 2}} (\frac{1}{1 + \frac{1}{e}}) = \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e + 1} . \end{matrix} \end{matrix}

(211)

Therefore, the function

R_{m}^{+} (x)

is satisfied as follows:

\begin{matrix} R_{m}^{+} (x) \leq lim_{N \to \infty} B_{2} (x) = \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e + 1} . \end{matrix}

(212)

where

m > 1

. For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} k_{f} (x) \leq \frac{1}{x} (2 + log (x)) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e + 1} . \end{matrix}

(213)

By the same method,

R_{m}^{+}

is approximated as follows:

\begin{matrix} R_{m}^{+} (x) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e + 1} . \end{matrix}

(214)

where

m > 1

. For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} \begin{matrix} k_{f} (x) & \leq \frac{1}{x} (2 + log (x)) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e + 1} . \end{matrix} \end{matrix}

(215)

(The end of the proof

R_{m}^{+} (x)

on Lemma )

The proof of

R_{m}^{-} (x)

is similar. According to (202), the following are satisfied:

\begin{matrix} \begin{matrix} R_{m}^{-} (x) & = lim_{N \to \infty} \sum_{n = 1}^{N} |{(Q_{f} (x))}^{(n)}| \leq lim_{N \to \infty} \sum_{n = 1}^{N} \frac{1}{e^{(n - 1)} {(x - 1)}^{1 / 2}} . \end{matrix} \end{matrix}

(216)

Put the function

A_{2} (x)

as follows:

\begin{matrix} A_{2} (x) = \sum_{n = 1}^{N} |\frac{{(- 1)}^{n - 1}}{e^{(n - 1)} {(x - 1)}^{1 / 2}}| . \end{matrix}

(217)

Hence, we can describe as follows:

\begin{matrix} A_{2} (x) = \frac{1}{e^{0} {(x - 1)}^{1 / 2}} + \frac{1}{e^{1} {(x - 1)}^{1 / 2}} + \dots + |\frac{{(- 1)}^{N - 1}}{e^{N - 1} {(x - 1)}^{1 / 2}}|, \end{matrix}

(218)

\begin{matrix} \frac{1}{e} A_{2} (x) = \frac{1}{e^{1} {(x - 1)}^{1 / 2}} + \frac{1}{e^{2} {(x - 1)}^{1 / 2}} + \dots + |\frac{{(- 1)}^{N - 1}}{e^{N} {(x - 1)}^{1 / 2}}| \end{matrix}

(219)

Subtract from (219) the equivalent (218), the following equivalent are satisfied:

\begin{matrix} \begin{matrix} (1 - \frac{1}{e}) A_{2} (x) & = lim_{N \to \infty} (\frac{1}{{(x - 1)}^{1 / 2}} - \frac{{(- 1)}^{N}}{e^{N} {(x - 1)}^{1 / 2}}) = \frac{1}{{(x - 1)}^{1 / 2}} . \end{matrix} \end{matrix}

(220)

Therefore, it is satisfied as follows:

\begin{matrix} \begin{matrix} lim_{N \to \infty} A_{2} (x) & = \frac{1}{{(x - 1)}^{1 / 2}} (\frac{1}{1 - \frac{1}{e}}) = \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e - 1} . \end{matrix} \end{matrix}

(221)

Therefore, the function

R_{m}^{-} (x)

is satisfied as follows:

\begin{matrix} R_{m}^{-} (x) \leq lim_{N \to \infty} A_{2} (x) = \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e - 1} . \end{matrix}

(222)

where

m > 1

. For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} k_{f} (x) \leq \frac{1}{x} (2 + log (x)) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e - 1} . \end{matrix}

(223)

By the same method,

R_{m}^{-}

is approximated as follows:

\begin{matrix} R_{m}^{-} (x) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e - 1} . \end{matrix}

(224)

where

m > 1

. For all sufficiently large

x > 1

, the following conditions are satisfied:

\begin{matrix} \begin{matrix} k_{f} (x) & \leq \frac{1}{x} (2 + log (x)) \leq \frac{1}{{(x - 1)}^{1 / 2}} \frac{e}{e - 1} . \end{matrix} \end{matrix}

(225)

(The end of the proof

R_{m}^{-} (x)

on Lemma ) □

Acknowledgments

We would like to thank everyone who supported this challenge and deeply respect the ideas they gave us.

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Entropy and Its Application to Number Theory

Abstract

1. Introduction

1.1.

1.2.

1.3.

1.4.

2. The Boltzmann Principle and the Planck Distribution Function

Introduction for Entropy S and the Planck Distribution Function

3. Expansion of the Planck Distribution Function

3.1. Entropy S π f that Partitioned by Approximation of π ( x )

3.2. Derivation of the Functions R α ±

3.3. The Expanded Planck Distribution Function n ± ( x , α )

3.4. Correspondence the Planck Distribution Function n ¯ ( ν , β ) and the Expanded Planck Distribution Function n ± ( x , α )

3.5. Fine Structure Constants

4. Application the Function R α ± to Number Theory

4.1. Examples Using the Function R α + for Deriving Von Koch’s Inequality

4.2. Example Using the Function R α − for the abc Conjecture

4.3. Conclusion and Application to Number Theory

5. Generalization of Entropy and Application to Dynamical Systems

5.1. About S π f ′ ( x ) and S π f ′ ′ ( x )

5.2. Generalization of Entropy Acceleration S D ′ ′ ( x )

5.3. Relationship between S D ′ ′ ( x ) and S π f ′ ′ ( x )

5.4. Logistic Function of Bacterial, Population Growth and Generalize Entropy

5.5. Complex Dynamic Systems.

5.6. Conclusions

6. The proof of Lemma 3.6 and Lemma 3.7

6.1. The proof of Lemma 3.6

6.2. The Proof of Lemma 3.7

Acknowledgments

References

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3.1. Entropy $S_{π_{f}}$ that Partitioned by Approximation of $π (x)$

3.2. Derivation of the Functions $R_{α}^{\pm}$

3.3. The Expanded Planck Distribution Function $n^{\pm} (x, α)$

3.4. Correspondence the Planck Distribution Function $\bar{n} (ν, β)$ and the Expanded Planck Distribution Function $n^{\pm} (x, α)$

4. Application the Function $R_{α}^{\pm}$ to Number Theory

4.1. Examples Using the Function $R_{α}^{+}$ for Deriving Von Koch’s Inequality

4.2. Example Using the Function $R_{α}^{-}$ for the abc Conjecture

5.1. About $S_{π_{f}}^{'} (x)$ and $S_{π_{f}}^{''} (x)$

5.2. Generalization of Entropy Acceleration $S_{D}^{''} (x)$

5.3. Relationship between $S_{D}^{''} (x)$ and $S_{π_{f}}^{''} (x)$