On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \binom{s+n-1}{n}\zeta(s+n). $$ I wonder whether variations of this identity also exist. For instance, are there similar binomial sums for $$(1-a^{1-s})\zeta(s) $$ for $a \in \mathbb{Z}\setminus\{2\}$, or is there something special about $a=2$ that makes it work?

And what about products like $$\zeta(s) \prod_{k=1}^{p} (1-a_{k}^{1-s})$$ for some sequence $a_{1}, \dots, a_{p} \in \mathbb{Z}$, does that expression equal any binomial sum(s) in terms of values of the Riemann zeta function?

N.B. I've also asked this question on MSE.

I've corrected some typos. The $2^{-s}$, $a^{-s}$, and $a_{k}^{-s}$ factors should have been $2^{1-s}$, $a^{1-s}$, and $a_{k}^{1-s}$, respectively.