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Consider the following two formulas for $\zeta(s)$ $$\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \binom{n}{k} \frac{(-1)^...

A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$ Many more identities can be found in articles by e.g. Borwein and Adamchik &...

On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...

Let $h(s,n)$ be: $$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$ and let $g(s,n)$ be: $$g(s,n)=\lim_{c\...

I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though. Numerical evidence suggests that: $$\lim_{N \to +\infty} \sum_{n=1}^N\...