# 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.

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