# First motivation for extending Riemann Zeta to complex domain?

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.

• 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. May 16, 2021 at 2:28
• 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. May 16, 2021 at 17:38
• 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.
– KCd
May 17, 2021 at 2:28
• 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).
– KCd
May 28, 2022 at 1:29