Can you provide a proof for at least one of the claims given below?
It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta function (Vardi 1991) . Similarly we can formulate the following claims:
Claim 1. $$\frac{\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{2^n-1}{3^n} \cdot \zeta(n+1)$$
The SageMath cell that demonstrates this claim can be found here.
Claim 2. $$\sqrt{3} \pi=\displaystyle\sum_{n=1}^{\infty}\frac{5^n-1}{6^n} \cdot \zeta(n+1)$$
The SageMath cell that demonstrates this claim can be found here.
Claim 3. $$(\sqrt{2}+1) \pi=\displaystyle\sum_{n=1}^{\infty}\frac{7^n-1}{8^n} \cdot \zeta(n+1)$$
The SageMath cell that demonstrates this claim can be found here.
Claim 4. $$(\sqrt{3}+2) \pi=\displaystyle\sum_{n=1}^{\infty}\frac{11^n-1}{12^n} \cdot \zeta(n+1)$$
The SageMath cell that demonstrates this claim can be found here.
