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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.

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Yes, the reference is to Euler's Introductio in Analysin Infinitorum (Introduction to Analysis of the Infinite) in which Euler investigates infinite series. The reference seems to point to Volume 1, Chapter XV, which is titled "Concerning Series Arising From the Expansion of Factors."

The English translation by Ian Bruce of the Introductio is in the public domain, and I'm guessing that you have found that file. If not, the full file can be found on Ian Bruce's site, 17centurymaths.com (or just click here for Chapter XV). The identity is specifically found on pages 470 and 471 of that translation, §273.

The identity is first explicity stated with $s = 1$ in §273. §274, however, is where he states the more generalised identity for case where $s \neq 1$ (referring to $s$ as $n$ instead), investigating this further in subsequent sections. Be warned, however, that Euler's notation, even post-translation, is very different from the modern standard (e.g. using simply $l$ sans parentheses to denote the natural logarithm function).

I hope this helps!

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  • $\begingroup$ I also found the same thing. I know he dealt with the case $s>1$, $s \in \mathbb{N}$. But I am really curious to know whether he really dealt with the case $s>1$, $s \in \mathbb{R}, s \not \in \mathbb{N}$. $\endgroup$ – math is fun Jun 16 at 16:22
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    $\begingroup$ As far as I can tell, he only examined $s \in \mathbb{N}$ in the Introductio but I'm not familiar enough with his other works to know if he ventured into $s \in \mathbb{R}$. $\endgroup$ – Hal Jun 16 at 17:44
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    $\begingroup$ Euler only considered the zeta function at integral values of $s$. He did not work with it as a function of a real variable. $\endgroup$ – KCd Jun 18 at 16:14

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