16. Riemann zeta function and quantum theory

The content of the article below is slightly different from the most recent article.Riemann zeta function and quantum theory, About roots of Riemann zeta function

 

Equation (1) from the previous article, Prime number and quantum mechanics, can be modified as follows.

 

$\beta^2=(1 \pm \sqrt{1-1/\theta^2})/2$

 

Here, we will focus on cases where $\theta$ is close to $1$. If it is less than $1$, it appears as an imaginary number, and if it is larger than $1$, it appears as a real number. This point appears to represent the boundary between atoms and elementary particles such as quarks. Also, if you change from $1/\theta$ to $\theta$, the opposite state appears. In other words, the potential appears as a reciprocal at this boundary. Real numbers viewed from outside the boundary appear as the reciprocal of the imaginary potential, and values calculated as real numbers inside the boundary appear as the reciprocal of the imaginary potential outside.

 

if $\theta \approx 1$ and $\cos^2\pi\theta \approx \theta^2$,

 

$\beta^2= (1 \pm i\tan\pi\theta)/2$

 

$=(\cos\pi\theta \pm i\sin\pi\theta)/2\cos\pi\theta$.

 

$\beta^{2s}= (\cos\pi s\theta \pm i\sin\pi s\theta)/2^s\cos^s\pi\theta$

 

It seems that approximating $\cos^2\pi\theta$ to $\theta^2$ is not just meant to be a convenient solution. It can be approximated by $\theta \approx \sin(\pi\theta/2)$. It seems like a natural phenomenon that occurs in real nonlinear differential equations. I can't think of a suitable example in fields other than quantum mechanics, but it is a case where the solution to an equation is derived completely differently depending on the scale of the variable. And this process seems to represent a middle ground between stochastic indeterminism and deterministic dynamics. According to the previous article,¹ $1/\beta$ is related with prime number. Considering that the complex function was created during the approximation and the case that $\theta$ is greater than $1$, we can set $\theta^2=\csc^2\pi\theta$.

 

$\beta^2= (1 \pm \cos\pi\theta)/2=\sin^2(\pi\theta/2), \cos^2(\pi\theta/2)$

 

Therefore, the Riemann zeta function can be roughly expressed as follows.

 

$\zeta(s)=\Sigma{\beta^{s}_i}=\Sigma{\cos^s(\pi\theta_i/2)}$

 

If we consider that the $\theta$ change between prime numbers is tiny and express it in integral form,

 

$\zeta(s) \approx \int{\cos^s(\pi\theta/2)}d\theta = \int{(\ln(r/r_0))^s}d(r/r_0)$

 

Here, in order to obtain a value similar to the actual zeta function, a way to reduce the differentiation interval needs to be found. In the case of $\theta$, a natural number can be expressed as $1/\cos\theta$ as well as a prime number, so the difference between the natural numbers is $\Delta (1/\cos\theta)$, and zeta function becomes as follows.

$\zeta(s) = \int{\cos^{s-2}\theta\sin\theta}d\theta=\cos^{s-1}\theta/(s-1)$

Because $\zeta(1)$ is almost proportional to $\ln (1/\cos\theta)$, the modified differential operator must be used for integration. Substituting $\Delta(1/\cos\theta)$ with $-\Delta(e^{-p/\cos\theta})$, 

where it is assumed that the interval will increase exponentially and $p$ is constant.

 

(1)  $\zeta(s) = p\int{\cos^{s-2}\theta\sin\theta e^{-p/\cos\theta}}d\theta$

Therefore, we can conjecture that equation (1) is the integral form of the zeta function. It is converted into another form as follows.

(2) $\zeta(s)=p\int x^{s-2} e^{-p/x}dx$,² $|x| \leq 1$

 

$\zeta(s)=p\int x^{-s} e^{-px}dx$, $|x| \geq 1$.

 

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^{1. Refer to Quantized time and Distance.}

^{2. The constant $p$ is assumed for convenience and does not appear to be a constant in reality. In physics, this appears to represent the procedure of creating mass and charge.}

 ^{2.a. If this assumption is correct and $p$ is a function of $s$ and almost same until $s$ becomes $s+1$, the following relationship is established, and a relationship of the form $p_s=f(s)$ can be obtained through comparison with the previously known $\zeta(s)$ value, $p_s(e^{-p_s}-\zeta(s)) = s\zeta(s+1)$. If it is changed to a sequence form rather than an integral form, $p_se^{-p_s} $ is thought to be related with the Bernoulli sequence.}

 ^{2.b. $\zeta(s)$ can also be expressed as the complex trigonometric function shown in the beginning of this article, $\beta^{2s}= (\cos\pi s\theta \pm i\sin\pi s\theta)/2^s\cos^s\pi\theta$. But here, it seems that the prime number is not $1/\beta$, but $\beta^2$ itself. A prime number is expressed in the form of both $1/\beta$ and $e^{1/\beta}$ as a real number. Assuming $1/\beta$ is a number, the interval between prime numbers is proportional to $e^{1/\beta}$, and assuming $e^{1/\beta}$ is a prime number, the interval is proportional to $1/\beta$ in the integral function. Therefore, it is assumed that $s$ is related to prime number and $\beta$. It seems to be saying that the real part of $s$ is $1/2$.}

 ^{* complex prime number: If $p_c$ is a complex prime number, $p_c=a+bi$, then $|p_c|=\sqrt{a^2+b^2}$ ($a, b$: real number).}

^{Last modified in 2023.12.17}