Nt.number theory – Accelerate the convergence of a product by multiplying by zeta values: historical?

Let $ R (s_1, dotsc, s_n) = prod_p r (p ^ {- s_1}, dotsc, p ^ {- s_n}) $, or
$ r $ is a rational function on $ n $ variables. Let's say we want to calculate the value of $ R (s_1, dotsc, s_n) $ for a choice of $ s_1, dotsc, s_n $ for which the infinite product converges. As it is well known and easy to see, we can dramatically speed up convergence speed (and dramatically reduce computational time for a desired level of precision) by multiplying or dividing everything first. $ R (s_1, dotsc, s_n) $ by some well chosen values ​​of the $ zeta $ a function.

When was this observation made for the first time, at least for some $ R $? I have heard of Gauss and / or Ramanujan, but it would be good to have a real reference. A friend assumes that Rosser and Schoenfeld had to do this in order to calculate some constants in the 1960s, but, again, I have no reference and would need it.