Entropy and Its Application to Number Theory

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Entropy and Its Application to Number Theory

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Seiji Fujino^*

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Submitted:

16 November 2023

Posted:

17 November 2023

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Abstract

In this paper, we propose an extension of the Planck distribution function derived from the Boltzmann principle. That is, we consider extending Planck's law with new distribution functions. In addition, using ideas applied to the expansion of the Planck distribution function, Von Koch's inequality is derived without using the Riemann hypothesis, showing that the abc conjecture is negated. We also describe some challenges for the future. Namely, we will discuss that Entropy is related to dynamical systems described by logistic function models, such as the bacterial and the population growth.

Keywords:

Subject: Physical Sciences - Thermodynamics

1. Introduction.

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

1.1

First, we explain the principle of Boltzmann’s Entropy S and the Planck distribution function to aid understanding. The Planck distribution function partitions particles P into resonators N and applies this partitioning method to Entropy S. Furthermore, this Entropy S is made to correspond to the average energy of resonators U and an energy element

ε

. In addition, The Planck distribution function is derived by differentiating with the average energy of resonators U.

1.2

Second, we describe that the expansion of the Planck distribution function which is main contents of 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 let x be a positive real number. The function

x / log (x)

is an approximation of the number of prime numbers

π (x)

. The function

R_{α}^{\pm} (x)

is defined and describe the relation between

S_{π_{f}} (x)

and

R_{α}^{\pm} (x)

. Furthermore, we attempt to compare the possibility of expanding the Planck distribution function by using the function

R_{α}^{\pm} (x)

. Besides, the relation between the constant

α

of the function

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. Von Koch’s inequality is derived without using the Riemann hypothesis. Namely, it proves the Riemann Hypothesis is correct. Additionally, we verify that the negation of the abc conjecture is 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 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.

2.1. 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 & : T h e 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}

(2.1)

□

Using the definitions above, the following equations are satisfied :

\begin{matrix} U_{N} = N U = P ε, \end{matrix}

(2.2)

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

(2.3)

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

(2.4)

where the inequality

P > N

is satisfied.

The concept of the Planck distribution is that the number of particles P is partitioned by the number of resonators N. Namely, the number of particles P is partitioned by the number of partitions

N - 1

. The number of particles P and resonators N can be regarded positive integer numbers. Therefore, we define the number of states

W_{N, P}

and Entropy (the Boltzmann Principle) S as follows :

Definition 2.2.

Let the number of particles P and the number of resonators N be positive integers.

\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}

(2.5)

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

(2.6)

\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}

(2.7)

□

Using the Stirling’s formula, for sufficiently large natural numbers P and N, the following conditions 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}

(2.8)

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 + P) log (N + P) - N log N - P log P\} \\ = k_{B} N \{(\frac{P}{N}) log N + (1 + \frac{P}{N}) log (1 + \frac{P}{N}) - \frac{P}{N} log P\} \\ = k_{B} N \{(1 + \frac{P}{N}) log (1 + \frac{P}{N}) - \frac{P}{N} log \frac{P}{N}\} . \end{matrix} \end{matrix}

(2.9)

Using the definition above, the equality (2.3)

P / N = U / ε

and (2.7)

S = S_{N, P} / N

, the equality (2.9) 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}

(2.10)

Differentiate both sides of the equation (2.10) 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}

(2.11)

Furthermore, differentiate both sides of the equation (2.11) 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}

(2.12)

The rate of change of Entropy

d S

is the multiplication of the rate of change of Energy U and the reciprocal of temperature T. Namely, the following relation between Entropy S, average energy per resonator U and Temperature T are satisfied :

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

(2.13)

Thus, using the equation (2.12) and (2.13), 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}

(2.14)

Integrating both sides of the equation (2.14) 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}

(2.15)

Here, put

ε

as follows :

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

(2.16)

Therefore, the following equations are satisfied :

\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}

(2.17)

The equation above (2.17) is determined the expression for the average energy of particles in a single mode of frequency

ν

in thermal equilibrium T, that is, called 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}, (t h e 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}

(2.18)

This is expressed the mean particle occupation number in thermal equilibrium T . This is called the Planck distribution function on this paper. Moreover, the equation (2.18) is transformed as follows :

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

(2.19)

The function

exp (- h ν β)

is called 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}

(2.20)

3. Expansion of the Planck distribution function.

3.1. Entropy $S_{π_{f}}$ .

We will continue the discussion with reference to ideas in subsection2.1. The number of particles P is replaced to the positive real number x. 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 the function a positive real number x by logarithm

log (x)

, that is, the function

x / log (x)

. This function

x / log (x)

is an approximation to the prime number

π (x)

. We show the function

R_{α}^{\pm} (x)

is derived as follows. First, we start with some definitions.

Definition 3.1.

Let

x > 1

be a positive real number, 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}

(3.1)

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

(3.2)

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

(3.3)

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}

. Therefore, 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 devide by Entropy

S_{π_{f}, x}

π_{f} (x)

as follows:

Definition 3.2.

Entropy

S_{π_{f}} (x) d i v i d e d b y π_{f} (x)

Let

x > 1

be a positive real number.

\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}

(3.4)

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

(3.5)

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

(3.6)

□

Note: Since the definition of Combination below formula (3.7) cannot define real values well, therefore, we adopted the definition of formula (3.4) using Stirling’s approximation.

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

(3.7)

In discussion below, 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}

(3.8)

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}

(3.9)

By the definition 3.2, for sufficiently large

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}

(3.10)

\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}

(3.11)

Using the function

Q_{f} (x)

above, the function

S_{π_{f}} (x)

under

π_{f}

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}

(3.12)

Differentiating Entropy

S_{π_{f}} (x)

under

π_{f}

as follows :

\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}

(3.13)

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}

(3.14)

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}

(3.15)

\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}

(3.16)

Repeating differential of the part of

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

on (3.16), 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}

(3.17)

\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}

(3.18)

Therefore, for all sufficiently large

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}

(3.19)

\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}

(3.20)

Furthermore, for all sufficiently large

x > 1

, the following are satisfied :

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

(3.21)

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

(3.22)

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 number, and

f (x)

be a real valued function. The function

k_{f} (x)

is defined as follows :

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

(3.23)

Namely, 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}

(3.24)

□

Let us call this function

k_{f} (x)

the Boltzmann variable function in the function

f (x)

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}

(3.25)

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}

(3.26)

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}

(3.27)

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}

(3.28)

□

The function

R_{m}^{+} (x)

is called an m-th absolute lower bound approximation of the Boltzmann variable function

k_{f} (x)

in the function

f (x)

. Similarly, the function

R_{m}^{-} (x)

is called an m-th lower bound approximation of the Boltzmann variable function

k_{f} (x)

in the function

f (x)

. Using the defintion above, the following inequality is satisdied :

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

(3.29)

where the function

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

represents the n-th derivation of

log (x)

. Moreover, the function

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

and

{log}^{n} (x)

to the n-th power represents

log (x)

Using equivalent (3.24), for all sufficiently large

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}

(3.30)

where the function

Q_{f} (x)

log (x)

. Because, by the equation(3.16),

\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}

(3.31)

Therefore, for sufficiently large

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}

(3.32)

Furthermore, 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}

(3.33)

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}

(3.34)

First order differentiation of Entropy

S_{π_{f}} (x)

is always positive values, that is

S_{π_{f}}^{'} (x) > 0

. Moreover, 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 $R_{α}^{\pm} (x)$ .

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 number

x > 1

, the function

R_{α}^{+} (x)

and

R_{α}^{-} (x)

are defined as follows :

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

(3.35)

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

(3.36)

The function

R_{α}^{\pm} (x)

are combined

R_{α}^{+} (x)

and

R_{α}^{-} (x)

as follows :

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

(3.37)

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}

(3.38)

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

(3.39)

□

This function

R_{α}^{\pm} (x)

is called an ±lower bound approximation of the Boltzmann variable function

k_{f} (x)

in the function

f (x)

and the constant

α

The relations of functions

R_{α}^{\pm} (x)

R_{m}^{+} (x)

and

R_{m}^{-} (x)

are satisfied as follows :

Lemma 3.6.

The relation

R_{m}^{+} (x) \geq R_{α}^{+} (x)

Let

α > 0

be a positve real number. There is an integer

m \geq 1

such that for all sufficiently large

x > 1

, the following inequality is satisfied :

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

(3.40)

where a positive real number α is satisfied as follows :

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

(3.41)

that is, satisfied as follows:

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

(3.42)

Using same as discussion, the following conditions are satisfied :

Lemma 3.7.

The relation

R_{m}^{-} (x) \geq R_{α}^{-} (x)

Let

α > 0

be a positve real number. There is an integer

m \geq 1

such that for all sufficiently large

x > 1

, the following inequality is satisfied :

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

(3.43)

where a positive real number α is satisfied as follows :

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

(3.44)

that is, satisfied as follows:

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

(3.45)

Proof.

The proof of Lemma 3.6 and Lemma 3.7 are described the following the Section 6.1. □

Consequently, for sufficiently large real number

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

(3.46)

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

(3.47)

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}

(3.48)

The second derivative of Entropy

S_{π_{f}}^{''} (x)

is suppressed from the bottom side by formula. Besides, the second derivative of Entropy

S_{π_{f}}^{''} (x)

is suppressed from the upper side by formula as follows.

Lemma 3.8.

R_{m}^{+} (x) \geq R_{α}^{+} (x), a n d R_{m}^{-} (x) \geq R_{α}^{-} (x) .

For all sufficiently large

x > 1

and a positive integer

m > 1

, the following inequalities are satisfied :

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

(3.49)

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

(3.50)

Proof.

The proof of Lemma 3.8 are described the following the Section 6.2. □

On the next subsection, we discuss the meaning of inequalities above Lemma 3.6 and Lemma 3.7.

3.3. The Expansion of the Planck distribution $n^{\pm} (x, α)$ by using $R_{α}^{\pm} (x)$ .

Next, we examine to define the distribution functions

n^{\pm} (x, α)

by using

R_{α}^{\pm} (x)

. Beside, integrate the inequality (3.46) and (3.47) as a variable x, Therefore, the following inequality is satisfied :

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

(3.51)

As the following equation is satisfied :

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

(3.52)

Hence, 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}

(3.53)

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}

(3.54)

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}

(3.55)

Thus, using inequality (3.53), the constant C is satisfied as follows :

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

(3.56)

Therefore, the inequality (3.53) is satisfied as follows :

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

(3.57)

For sufficiently large positive real number

x > 1

, the function

S_{π_{f}}^{'} (x)

is satisfied as follows :

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

(3.58)

According to inequalities (3.57) and (3.58),

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

(3.59)

Therefore, by (3.59) the following inequality is derived :

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

(3.60)

Focusing on equality of the inequality (3.60), we define the distribution function

n^{\pm} (x, α)

as follows:

Definition 3.9.

The distribution functions

n^{\pm} (x, α)

are defined as follows:

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

(3.61)

where

α > 0

. □

The definition above 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}

(3.62)

Thus, this distribution function

n^{\pm} (x, α)

is regards as the approximate density of prime numbers

x / π (x)

until the number x. Besides, this function

n^{\pm} (x, α)

is regarded as one of the distribution functions. Furthermore, this function

n^{\pm} (x, α)

is seems to expand 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}

(3.63)

is called the expansion of Boltzmann factor or

R_{α}^{\pm}

factor . We will consider the further relationship in the next subsection.

3.4. Corresponding the Planck distribution function and the distribution function $n^{\pm} (x, α)$ .

We examine to correspond the Planck distribution function

\bar{n} (ν, β)

and the distribution function

n^{\pm} (x, α)

as follows :

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

(3.64)

where

\begin{matrix} \begin{matrix} h & : t h e P l a n c k c o n s t a n t, \\ ν & : F r e q u e n c y, \\ β & : I n v e r s e t e m p e r a t u r e . \end{matrix} \end{matrix}

(3.65)

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

exp (- h ν β)

\begin{matrix} h ν β \end{matrix}

(3.66)

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}

(3.67)

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}

(3.68)

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}

(3.69)

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

(3.70)

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)}{\overset{P : = x}{}} ↓ ↑ \overset{α : = \frac{e}{\sqrt{2 π} h}}{\overset{x : = \frac{\mp 1}{e (1 - ν β)}}{}} & \overset{N : = π_{f} (x)}{\overset{P : = x}{}} ↓ ↑ \overset{α : = \frac{e}{\sqrt{2 π} h}}{\overset{x : = \frac{\mp 1}{e (1 - ν β)}}{}} \\ 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}

(3.71)

Corresponding the above, the distribution function

n^{\pm} (x, α)

becomes to expand the Planck distribution function. Namely, the following conditions are satisfied :

Suggestion 3.10.

The expansion of the Planck distribution

\bar{n} (ν, β)

Let

α > 0

be a real number constant. For all real number

x > 1

the following equation is satisfied :

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

(3.72)

where

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

(3.73)

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(3.69) is satisfied as follows:

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

(3.74)

Moreover, according to the method to divide each S and

S_{π_{f}} (x)

, we remember that the following corresponds :

The number of particles P is replaced to the positive real number x.
The number of resonators N is replaced to approximate number of prime numbers $π (x) \sim \frac{x}{log (x)}$ .

Therefore, we consider the correspondence the between

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

(3.75)

and

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

(3.76)

Namely, we suppose the following correspondence is considered :

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

(3.77)

Thereby, We consider the correspondence the between Planck’s law U and the following function

U_{x, α}^{\pm}

Definition 3.11.

The real valued function

U_{x, α}^{\pm}

as the expansion of Planck’s law U.

Let

α > 0

be a real number constant. For sufficiently large real number

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{x}{log (x)} \\ = \frac{x}{log (x) (exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1)} \\ \sim \frac{π (x)}{exp (\frac{1}{x R_{α}^{\pm} (x)}) - 1} (α > 0) . \end{matrix} \end{matrix}

(3.78)

□

Using the suggestion3.10 and the definition3.11, the following suggestion is given:

Suggestion 3.12.

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 real numbers

x > 1

and

α > 0

such that the following equality is satisfied :

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

(3.79)

where the following conditions are satisfied :

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

(3.80)

Namely, the real valued function

U_{x, α}^{\pm}

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

According to the suggestion 3.12 above, under the condition that the product of the Planck constant h and the frequency

ν

, that is,

h ν

is an approximation of the number of prime numbers

π (x)

. Planck’s law U is seems to take discrete values and has an approximate spectrum of prime numbers. It is possible that the discrete values of an element of energy are related to the distribution of prime numbers.

3.5. A kind of fine-structure constant.

The distribution function

n^{\pm} (x, α)

and the Planck distribution

\bar{n} (ν, β)

is associated by a constant

α

. The constant

α = e / \sqrt{2 π} h

is thought like the fine-structure constant that associated with the Planck constant h.

Let a positive real number

α

set as follows :

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

(3.81)

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}

(3.82)

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}

(3.83)

Thus, the positive real number

α

is satisfied such that

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

(3.84)

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 3.13.

The ratio of the Boltzmann constant and the Planck constant.

Let

α > 0

be a positive real number (constant). For sufficiently large

x > 1

, the following formulas are satisfied :

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

(3.85)

Namely, the ratio of the Boltzmann constant

k_{B}

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

π (x)

until x. □

Using discussions above, a constant

α

can be associated between the Planck distribution function

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

(3.86)

and the expansion of the Planck distribution function

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

(3.87)

Namely, suppose that a constant

α > 0

is decided. Specially, 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}

(3.88)

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

(3.89)

Namely, The function

R_{α}^{\pm} (x)

is changed and depended by a constant

α > 0

. Therefore, the constant

h_{α}

can defined for each constant

α

By lemma3.10, Modern physics may be a special case that satisfy the following condition:

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

(3.90)

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

(3.91)

Therefore, the following suggestion is stated :

Suggestion 3.14.

Let

α > 0

be a positive real number (constant). The constant

h_{α}

can be selected as follows :

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

(3.92)

where the following inequality is satisfied :

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

(3.93)

Namely, the condition of the equality is satisfied as follows :

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

(3.94)

Therefore, 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 using the Planck constant as follows.

In this paper, we describe it as original the fine-structure constant

α_{p}

to distinguish it from the real number

α

. Besides, describe it as the elementary charge

e_{p}

to distinguish it from the Napier’s number e.

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

(3.95)

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}

(3.96)

Therefore, the relation the fine-structure constant

α_{p}

and the real number

α

in this paper 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}

(3.97)

On the following section, using the function

R_{α}^{\pm} (x)

, 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}}, \end{matrix}

(3.98)

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

(3.99)

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

(3.100)

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

(3.101)

4. Application the function $R_{α}^{\pm} (x)$ to Number Theory.

4.1. Examples using the function $R_{α}^{\pm} (x)$ for deriving Von Koch’s inequality.

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

α

and the function

R_{α}^{\pm} (x)

. (Refer to Fujino [20])

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 number

x \geq 2

, 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}

(4.1)

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}

(4.2)

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

(4.3)

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

(4.4)

□

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

and for all sufficiently large

x \geq 2

, 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}

(4.5)

Proof.

Using Theorem(4.1), put a positive real number

α > 0

as follows:

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

(4.6)

Therefore, the inequality (4.5) is satisfied.

□

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

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

(4.7)

Comparing inequalities(4.5) and (4.7), 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}

(4.8)

Namely, the asymptotic of (4.5) gives better than that of (4.7).

Therefore, let be

α = 2 / \sqrt{2 π}

(h_{α} = e / 2)

. the theorem above are 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

and 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}

(4.9)

Proof.

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

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

(4.10)

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}

(4.11)

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}

(4.12)

Thus, the positive real number

α > 0

is satisfied conditions of (4.10) and (4.11). 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}

(4.13)

□

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}

(4.14)

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}

(4.15)

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}

(4.16)

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}

(4.17)

□

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

(4.18)

where

C, ϵ

and x are real numbers. Namely,

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

(4.19)

Proof.

Using theorem(4.1), put a positive real number

α > 0

as follows:

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

(4.20)

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}

(4.21)

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}

(4.22)

□

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}

(4.23)

where

C, ϵ

and x are real numbers. Namely,

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

(4.24)

□

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

4.2. Example using the function $R_{α}^{\pm} (x)$ for the abc conjecture.

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

α

. Namely, using the constant

α

, the following theorems are satisfied : (Refer to Fujino [21])

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}

(4.25)

where the following equation is satisfied :

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

(4.26)

□

Set the constant

α

as follows:

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

(4.27)

Therefore, the following is satisfied :

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

(4.28)

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 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}

(4.29)

Namely, There is 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 negative 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}

(4.30)

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. We will investigate the relation Entropy and Number Theorem further.

4.3. Conclusion and Application to Number Theory.

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

π (x)

, that is, the function

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}

(4.31)

where the constant h is Planck constant.

Furthermore, using new distribution function

R_{α}^{\pm}

, we evaluated Von Koch’s inequality that equivalent to the Riemann Hypothesis and the abc conjecture.

Namely, we consider the different system 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}

(4.32)

Moreover, the abc conjecture is satisfied as follows :

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

(4.33)

Namely, we considered that there exist the relation between new distribution function

n^{\pm} (x, α)

and the Boltzmann principle, furthermore the Planck distribution function. Furthermore, it is meaningful that there exist the relation between statistical mechanics and Number theory.

In the future, based on the above measures, It may be possible that there exists new constant

h_{α}

and non-constant

k_{f} (x)

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

5. Generalization and application to dynamical systems.

We would describe some future issues on this section. Namely, we consider that that Entropy is related to dynamical systems described by logistic function models, such as bacterial and population growth.

5.1. What are $S_{π_{f}}^{'} (x)$ and $S_{π_{f}}^{''} (x)$ ?

We would like to consider what

S_{π_{f}}^{'} (x)

and

S_{π_{f}}^{''} (x)

are (Refer to Nicolis, Prigogine [18,19]). Let

x > 1

be a real number. 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}

(5.1)

where these functions 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}

(5.2)

The first derivative of function

S_{π_{f}} (x)

, that is,

S_{π_{f}}^{'} (x)

and the second 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}

(5.3)

where the function

k_{f} (x)

is regard as a function decided by a real number x and the function

π_{f} (x)

. The function

Q_{f} (x)

can be regard as the position divided a real number x by

Q_{f} (x)

. The first derivative of

Q_{f} (x)

, that is,

Q_{f}^{'} (x)

can be regard as the slope of the function

Q_{f} (x)

(the change

Q_{f}^{'} (x)

of the position

f (x)

, the charge or potential

Q_{f} (x)

of the position

f (x)

). Entropy velocity can be regraded as Entropy generation (Refer to Nicolis, Prigogine [18,19]). We consider the generalization below subsection.

5.2. Generalize of the function $S_{D}^{''} (x)$ .

We generalize the equation above (3.51) as follows. Let

x > 1

be real number 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}

(5.4)

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

(5.5)

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

(5.6)

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}

(5.7)

Moreover, the function

k_{D} (x)

can be regard as the function decided by x,

ξ

and

D (x)

. The function

Q_{D} (x)

can be regard as the position divided a real value

ξ x

Q_{D} (x)

. The first 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. New distribution function $R_{α}^{\pm} (x)$ .

We examine that the correspondence between generalized the equation(5.6) of

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

and the equation(3.51) of

S_{π_{f}}^{''} (x)

. We put 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}

(5.8)

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}

(5.9)

This equation (5.9), that is (3.51), can be regarded as an application of the equation (5.6). In the subsections below, we examine some laws can be regarded as the accelerations of

S_{D} (x)

, that is (5.6).

5.4. Logistic function of the bacterial and the population growth.

We examine the relationship between generalized the function

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

of equation (5.6) and the logistic function such that the bacterial and the population growth.

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

t > 0

, let

N (t)

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

\begin{matrix} \begin{matrix} S_{D}^{''} (x) & : = - r (< 0), \\ 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}

(5.10)

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) & : t h e b a c t e r i a l o r t h e 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}

(5.11)

Thus, the equation(5.6) 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}

(5.12)

Thus, 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}

(5.13)

Therefore, the following equation is obtained :

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

(5.14)

The equation(5.14) above is the logistic function of dynamics. In other words, the equation (5.14) derived from the equation(5.6) can be regarded as an application of the dynamical system. Therefore, we consider that Entropy and dynamical system are closely related and are studying these applications.

5.5. Conclusion.

We considered the following possibilities :

for sufficiently large

x > 1

and a constant

ξ > 0

, the equation (5.6)

\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 and approximate representation of the equation (5.14)

\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.

According to partition the Boltzmann principle by the number of prime numbers, Entropy

S_{π_{f}} (x)

(the second law of

π_{f}

) can be related to Quantum mechanics(statistical mechanics) and Number theory.

Besides, Entropy acceleration

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

(the second derivative of Entropy

S_{D} (x)

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

Furthermore, according to the theory of quantum mechanics, an atom can only take discrete spectral values. Similarly, the placement of planetary systems such as the solar system, galaxies, and clusters of galaxies can be considered to only take discrete spectral values. This discrete spectral arrangements seems to be related to Entropy.

Entropy was related to quantum mechanics by Boltzmann and Planck. Prigogine and Nicolis associated Entropy with complex systems. We further developed these concept of Entropy, proposed an extension of the Planck distribution, and attempted to relate Entropy to dynamical systems and classical mechanics. In addition we applied the expansion of Entropy to Number theory.

In the future, We would investigate why these relationships appear when the partitions number of the Boltzmann principle is partition by the number of prime numbers. We hope 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 [22])

6. The proof of Lemma 3.6 , Lemma 3.7 and Lemma 3.8

6.1. The proof of Lemma 3.6 and Lemma 3.7

Proof.

Let n

\geq 1

a positive integer. For all sufficiently large positive real number

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}

(6.1)

Therefore, dividing the end of the formula(6.1), 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}

(6.2)

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}

(6.3)

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}

(6.4)

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}

(6.5)

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}

(6.6)

Therefore,

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

(6.7)

\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}}{e^{N} x^{(N + 1)}} . \end{matrix}

(6.8)

Add the equation(6.7) and (6.8). 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}}{e^{N} x^{(N + 1)}}) = \frac{1}{x} . \end{matrix}

(6.9)

Therefore,

\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}

(6.10)

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}

(6.11)

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}

(6.12)

Put the positive real number

α > 0

such that as follows:

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

(6.13)

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}

(6.14)

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

(6.15)

Furthermore, 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}

(6.16)

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

(6.17)

where

α > 0

is a real number. (The end of the proof of Lemma3.6)

By the same method, for integer

m > 1

, the function

R_{m}^{-}

is approximated as follows:

\begin{matrix} \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}

(6.18)

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

(6.19)

For all

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}

(6.20)

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

(6.21)

(The end of the proof of Lemma3.7). □

6.2. The proof of Lemma 3.8

Proof.

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}

(6.22)

Therefore, the following inequality are satisfied :

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

(6.23)

Furthermore, the inequality

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

(6.24)

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}

(6.25)

\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}

(6.26)

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}

(6.27)

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}

(6.28)

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}} + \frac{1}{e^{2} {(x - 1)}^{1 / 2}} - \dots, \end{matrix}

(6.29)

\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}}{e^{N} {(x - 1)}^{1 / 2}} \end{matrix}

(6.30)

Add the equivalent(6.29) and (6.30), 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}

(6.31)

Therefore,

\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}

(6.32)

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}

(6.33)

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}

(6.34)

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}

(6.35)

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}

(6.36)

The proof of

R_{m}^{-} (x)

is similar. □

Acknowledgments

We would like to thank all the people who supported this challenge and to express my deepest respect for giving us the idea. In the future, I would take on the challenge of discussing the relationship between entropy and the inverse square law, such as Newton’s law in classical gravity theory and Coulomb’s law in electromagnetism.

References

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P.L.Chebychev. Sur la totalité des nombres premiers inf érieursáune limite donneé, J. de Math. Pures Appl., 17:341–365, 1852.
Max Planck, Vorlesungen über die Theorie der Wärmestrahlung, J.A Barth, 1906.
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math. 41: 119-196 (1916).
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André LeClair, Riemann Hypothesis and Random Walks: The Zeta Case ,Symmetry 2021, 13, 2014.
Von Koch H, Sur la distribution des nombres premiers, Acta Mathematica, 24 159–182, 1901.
Entropy of Riemann zeta zero sequence, Advanced Modeling and Optimization, Volume 15, Number 2,O.Shanker, 2013.
Ioannis Kontoyiannis, Counting the Primes Using Entropy, The 2008 IEEE Information Theory Workshop, Porto, Portugal, Lecture given on Thursday, May 8 2008.
Arturo Ortiz, Hans Henrik Støleum. Studies of entropy measures concerning the gaps of prime numbers, arXiv:1606.08293 [math.GM], 2016.
Michael Atiyah. The Riemann Hypothesis. 2018.
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Michel Waldschmidt, On the abc Conjecture and some of its consequences,Université P. et M. Curie (Paris VI), 2018.
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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.

2.1. Introduction for Entropy S and the Planck distribution function.

3. Expansion of the Planck distribution function.

3.1. Entropy S π f .

3.2. Derivation of the functions R α ± ( x ) .

3.3. The Expansion of the Planck distribution n ± ( x , α ) by using R α ± ( x ) .

3.4. Corresponding the Planck distribution function and the distribution function n ± ( x , α ) .

3.5. A kind of fine-structure constant.

4. Application the function R α ± ( x ) to Number Theory.

4.1. Examples using the function R α ± ( x ) for deriving Von Koch’s inequality.

4.2. Example using the function R α ± ( x ) for the abc conjecture.

4.3. Conclusion and Application to Number Theory.

5. Generalization and application to dynamical systems.

5.1. What are S π f ′ ( x ) and S π f ′ ′ ( x ) ?

5.2. Generalize of the function S D ′ ′ ( x ) .

5.3. New distribution function R α ± ( x ) .

5.4. Logistic function of the bacterial and the population growth.

5.5. Conclusion.

6. The proof of Lemma 3.6 , Lemma 3.7 and Lemma 3.8

6.1. The proof of Lemma 3.6 and Lemma 3.7

6.2. The proof of Lemma 3.8

Acknowledgments

References

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3.1. Entropy $S_{π_{f}}$ .

3.2. Derivation of the functions $R_{α}^{\pm} (x)$ .

3.3. The Expansion of the Planck distribution $n^{\pm} (x, α)$ by using $R_{α}^{\pm} (x)$ .

3.4. Corresponding the Planck distribution function and the distribution function $n^{\pm} (x, α)$ .

4. Application the function $R_{α}^{\pm} (x)$ to Number Theory.

4.1. Examples using the function $R_{α}^{\pm} (x)$ for deriving Von Koch’s inequality.

4.2. Example using the function $R_{α}^{\pm} (x)$ for the abc conjecture.

5.1. What are $S_{π_{f}}^{'} (x)$ and $S_{π_{f}}^{''} (x)$ ?

5.2. Generalize of the function $S_{D}^{''} (x)$ .

5.3. New distribution function $R_{α}^{\pm} (x)$ .