26. Supplementary explanation of article 21.

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

 

This article is more detailed explanation about the relation between $b_1(x)$ and $b_0(x)$ in the previous article¹. 

 

(1)   $b_0(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n+1}(px)^n}{(n+1)(n+2)}$

 

($ b_0(x)=(\frac{x}{e^{px}-1})' \approx \sum_{n=0}^{\infty} \frac{(-1)^{n+1}(px)^{n}}{n!(n+1)(n+2)} $)

 

Here, $b_0(x)$ is a little different from the Riemann zeta function in the previous articles. Assuming that $b_0(x)$ always converges to $0$ as $x$ goes to $\infty$, the value of interest is when $x$ approaches $0$. Then, $b_0(x)$ can be modified as belows.²

 

$(xb_0(x))'=\sum \frac{(-1)^{n+1}(px)^n}{n+2}$

 

$(x^2(xb_0(x))')'=-1+px-(px)^2+(px)^3+...=-\frac{1}{px+1}$

 

$(xb_0(x))'=b_0(x)+xb'_0(x)= b_0(x)-xB(s)b_1(x) =-p(s)\frac{\ln(px+1)}{px^2}$

 

$B(s)$, $p(s)$ are assumed as functions of $s$. If all integration section other than the pole are $0$ and assuming that the same $b_n(x)$ can be used when $x$ approaches the pole, then both sides can be multiplied by $x^{s-2}$ and integrated to write as follows.

 

$\zeta(s)-B(s)\zeta_1(s+1)=-\frac{p(s)}{p}\int x^{s-4}\ln(px+1)dx=-p(s)c_{s-3}$

 

Because $p\zeta_1(s+1)=(s+1)\zeta(s+2)+1/2$ from the equation (2) in the article 21,

 

$\zeta(s)=-\frac{1}{2(s-1)}+\frac{p}{(s-1)B(s-2)}(\zeta(s-2)+p(s-2)c_{s-5})$

 

If $\zeta(s)$ is similar in the article 21, $B(s)=s/p$ and $p(s)$ must be the form of $p(s)c_{s-3}=a_{s}$. 

 

(2)  $\zeta(s)=-\frac{1}{2(s-1)}+\frac{p^2}{(s-1)(s-2)}(\zeta(s-2)+p(s-2)c_{s-5})$

 

(3)   $\zeta(2s)=-\frac{1}{2(2s-1)}+\frac{p^2}{(2s-1)(2s-2)}(\zeta(2s-2)+p(2s-2)c_{2s-5})$

 

Assuming $c_s=\frac{1}{p}\int x^{s-1}\ln(px+1)dx$, $c_s$ can be integrated when $s$ is a natural number.

 

$c_sp^s=\frac{{}_{s-1}C_0}{s^2}-\frac{{}_{s-1}C_1}{(s-1)^2}+...+(-1)^{s-1}$, $C$: combination

 

$s$ can be extended to complex number.

 

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[Added in 2024.01.23]

 

$b_0(x)$ can also be modified as follows.

 

$(b_0(1/x)/x)'=-\frac{1}{p}\sum \frac{(-p/x)^{n+2}}{n+2}$

 

$(b_0(1/x)/x)''=-\frac{1}{p^2}\sum (-p/x)^{n+3}=\frac{p}{x^3}(1-(p/x)+(p/x)^2-...)$

 

Substituting $1/x$ with $1/y-1$,

 

$(b_0(1/x)/x)''=-\frac{1}{p^2}\sum (-p(1/y-1))^{n+3}=\frac{p(y-1)^3}{y^2((1-p)y+p)}$ when $y -> 1$

 

$(b_0(1/x)/x)'=b'_0(1/x)/x-b_0(1/x)/x^2 = \int \frac{p(y-1)^3}{y^2((1-p)y+p)}dy$

 

(a)   $xb'_0(1/x)-b_0(1/x) = x^2\int \frac{p(y-1)^3}{y^2((1-p)y+p)}dy=-x^2\int \frac{pdx}{x^2(x+1)^3}$

 

$=p(3x-1 - x^2(\frac{3}{x+1}+ \frac{2}{(x+1)^2}+ \frac{1}{(x+1)^3} ))$

 

These relations may be used when $s<3$.

 

Functions that have an limiting value at a specific point but are not differentiable are modified in various ways near that value. In particular, it can be used to transform the function that is difficult to integrate.

 

After integrating the equation (a), it remains that $b_0(1/x)-xb'_0(1/x)=$(constant).

Therefore, the equation (2) becomes

 

(b)   $\zeta(s)+\frac{1}{2(s-1)}=\frac{p}{s-1}(\zeta(s-1)+p\frac{p(s)}{(s-2)})$

 

$p(s)$ can be conjectured like this.

 

(c)   $\frac{p^2}{(s-1)(s-2)}p(s)=-a_s+\frac{p}{s-1}a_{s-1}+\frac{1}{2(s-1)}$

 

When $1 \leq s < 3$, $s$ can be written like belows.³

 

(d)   $s-n=\cos^2\frac{\pi(s-n+1)}{2((s-n)!+1)}=\cos^2\frac{\pi\epsilon}{2}, |s-n|<1, n=1, 2$

 

The reason why $s$ is written in this way is to define factorial function for $s$ that is not a natural number.

 

Considering $\sin\frac{\epsilon\pi}{2} =1/\epsilon \approx 1$ for the certain $\epsilon$, the equation (d) can be modified differently. Therefore, if $a_s$ is correct, $p(s)$⁴ can be found using this relationship.

 

 

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^{1. Refer to About the roots of Riemann zeta function.}

^{2. Using the original $b_0(x)$, $(x^2(xb_0(x))')'=-e^{-px}, (xb_0(x))'=e^{-px}/px^2$. In this case, the relationship between $\zeta(s)$ can be expressed as the gamma function $\int x^{s-4}e^{-px}dx$.}

^{3. This is closely related to how $\zeta_n(s)=\int_{0}^{1} x^{s-2-n}b_n(x)dx $ will be expressed.}

^{4. It is referred to as $p(s)$ for convenience, but it does not mean the same function as $p(s)$ mentioned in the beginning of this article. Physically, this process appears to describe the formation of elementary particles and atoms depending on the value of $s$.}