In the following answer it has been claimed that "The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity for $s>1$ appears." Can anyone please tell me on which page can I find the following fact? I found a book by Euler, which is translated into English and it is titled as 'Introduction to Analysis of the Infinite ' Book 1. I found at page 235 of that book that Euler was aware of the fact that harmonic series, i.e. when $s=1$, is divergent. But I can't still find the case where $s>1$.
Here is Wilkins's translation of Riemann's paper. Remarks at the start of the paper show that Riemann was well aware of the convergence issues:"For this investigation my point of departure is provided by the observation of Euler that the product $\prod\frac{1}{1-\frac{1}{p^{s}}}=\sum\frac{1}{n^{s}}$ if one substitutes for $p$ all prime numbers, and for $n$ all whole numbers. The function of the complex variable $s$ which is represented by these two expressions, wherever they converge, I denote by $\zeta(s)$. Both expressions converge only when the real part of $s$ is greater than $1$; at the same time an expression for the function can easily be found which always remains valid". The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity for $s>1$ appears. Although Euler did not work with convergence in the modern sense he already knew the difference between $s=1$ and $s>1$ cases since he summed the Basel series in 1735 ($s=2$, later all even $s$), and used unlimited growth of harmonic series to prove infinitude of primes in that 1737 paper.