where $$C$$ is the value of the composite zeta function at $$2$$ and $$P$$ is the prime zeta function at $$2$$. My question is this. What will be the value of $$C$$?

Aidyn Meza
2022-09-20
Answered

Trying to calculate the value of $$\frac{{\pi}^{4}}{90}$$. Although I know the exact value (which I found on google to be $$\frac{{\pi}^{4}}{90}$$) but I wanted to derive it by myself. While doing so, I arrived at this rather peculiar expression: $C=\frac{7{\u213c}^{4}}{720}-\frac{1}{2}-\frac{P}{2}$

where $$C$$ is the value of the composite zeta function at $$2$$ and $$P$$ is the prime zeta function at $$2$$. My question is this. What will be the value of $$C$$?

where $$C$$ is the value of the composite zeta function at $$2$$ and $$P$$ is the prime zeta function at $$2$$. My question is this. What will be the value of $$C$$?

You can still ask an expert for help

Guadalupe Reid

Answered 2022-09-21
Author has **8** answers

Assuming

$$\zeta (s)=\sum _{n}\frac{1}{{n}^{s}}$$

$$P(s)=\sum _{p\text{prime}}\frac{1}{{p}^{s}}$$

$$\mathcal{C}(s)=\sum _{n\text{composite}}\frac{1}{{n}^{s}}$$

then

$$\mathcal{C}(s)=\zeta (s)-P(s)-1$$

and$$\mathcal{C}(2)=\zeta (2)-P(2)-1=\frac{{\pi}^{2}}{6}-P(2)-1\approx 0.192687$$

$$\zeta (s)=\sum _{n}\frac{1}{{n}^{s}}$$

$$P(s)=\sum _{p\text{prime}}\frac{1}{{p}^{s}}$$

$$\mathcal{C}(s)=\sum _{n\text{composite}}\frac{1}{{n}^{s}}$$

then

$$\mathcal{C}(s)=\zeta (s)-P(s)-1$$

and$$\mathcal{C}(2)=\zeta (2)-P(2)-1=\frac{{\pi}^{2}}{6}-P(2)-1\approx 0.192687$$

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