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Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime.

$$\zeta(s)=\sum_{n}\frac{1}{n^{s}}=\prod_{p}(1-\frac{1}{p^{s}})^{-1}$$

Riemann is considered to have been the first to extend the domain $s$ from the real numbers $\mathbb{R}$ to the complex domain $\mathbb{C}$.

Question 1: what motivated the need to widen the domain from the reals to the complex domain?

Question 2: what is the history around extending the domain from integers $s\in\mathbb{Z}$ to $s\in\mathbb{R}$?

For clarity, I am not asking about the analytic continuation of the zeta function.

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  • $\begingroup$ Whenever you have an ANALYTIC function, it is reasonable to study it in complex domain. (The reason is uniqueness theorem). So there is nothing special about zeta function here. $\endgroup$ May 16, 2021 at 2:28
  • $\begingroup$ For Question 2: Continuous (even analytic) functions on $\mathbb{R}$ are certainly better to study than functions $\mathbb{Z}\to \mathbb{R}$ especially when the range consists not only of integers. $\endgroup$
    – markvs
    May 16, 2021 at 17:38
  • $\begingroup$ Riemann’s motivation for viewing the zeta-function as a function of a complex variable was to outline a set of ideas based on complex analysis that he expected would lead to a proof of the prime number theorem (and they eventually did). This is the content of his paper on the zeta-function. Nobody before him had considered the zeta-function’s domain beyond the real numbers. $\endgroup$
    – KCd
    May 17, 2021 at 2:28
  • $\begingroup$ Euler only considered the zeta-function at integers, not as a function of a real variable. (His proof of infinitude of the primes was at argument at $s = 1$, not the more rigorous proof presented today using a limit as $s \to 1^+$.) I would not be surprised if the first person to seriously consider the zeta-function as a function of a real variable was Dirichlet as part of his proof of infinitude of primes in arithmetic progressions. He was using $L$-functions of characters of a real variable bigger than $1$, and this includes the trivial character (essentially the zeta-function). $\endgroup$
    – KCd
    May 28, 2022 at 1:29

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