Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?

Question

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?

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

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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.

Answer 1

First note that there is a typo in the formula you cite: it should be $$ (1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \binom{s+n-1}{n}\zeta(s+n) $$ ($1-s$, not $-s$). Something "special" in the number $2$ can be found, since $(1-2^{1-s})\zeta(s) = \eta(s)$ (Dirichlet eta function). However, the above formula can be generalised for a general integer $a \geq 2$. I will refer to the paper Lee, H.; Ok, B. M.; Choi, J. Notes on some identities involving the Riemann Zeta function (2002). Communications of the Korean Mathematical Society 17(1):165-173 for such identities.

First of all, we have the following identities obtained by Ramaswami:

$$ (1-3^{1-s})\zeta(s) = 1+2 \sum_{n=1}^{\infty} \frac{(s)_{2n}}{(2n)!} \zeta(2n+s) 3^{-2n-s} $$

$$ (1-2^{-s}-3^{-s}-6^{-s})\zeta(s) = 1 +2 \sum_{n=1}^{\infty} \frac{(s)_{2n}}{(2n)!} \zeta(2n+s) 6^{-2n-s}$$

The first one is the case $a=3$ of your first question, while the second one is a special case of the analogous of your second question, but with a sum instead of a product.

The general case for any integer $a \geq 2$ has been proven by Apostol. Actually, he obtained many interesting identities of this kind:

$$ (1-a^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \frac{(s)_{n} \zeta(n+s)}{n! a^{n+s}} \frac{B_{n+1}(a) - B_{n+1}}{n+1} $$

$$ (1-a^{1-s})\zeta(s) = \sum_{h=1}^{a-1} h^{-s} + \sum_{n=1}^{\infty} (-1)^n \frac{(s)_{n} \zeta(n+s)}{n! a^{n+s}} \frac{B_{n+1}(a) - B_{n+1}}{n+1} $$

$$ (1-a^{1-s})\zeta(s) = \frac{1}{2} \sum_{h=1}^{a-1} h^{-s} + \sum_{n=1}^{\infty} \frac{(s)_{2n} \zeta(2n+s)}{(2n)! a^{2n+s}} \frac{B_{2n+1}(a)}{2n+1} $$

I'm not aware of any formula of this kind for a product as the one of your second question.