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Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zet...
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 …