TITLE: Why does the Riemann zeta function have zeros in the complex plane? How is it possible to find them?
QUESTION [8 upvotes]: I ask this because, according to Euler's product formula, Riemann's zeta function =(1/something), so how could that be zero?
Also, how could one find zeros that are on the negative side and find a zero in the "critical strip", that is, when the real part of the input is between 0 and 1?

REPLY [2 votes]: Inferences about location of zeros or poles of a function from a product formula are valid only in a region where the product converges to the function.
The constant function $f(x)=1$ has no zeros or poles but can be written as a product
$$f(x) = (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})\dots$$
which superficially predicts zeros of $f(x)$, and poles of $1/f(x)$, at all $2^n$-th roots of unity.  The formula holds for $|x|<1$ so there is no contradiction.
For $\zeta(s)$ the non-trivial zeros are known to exist only through Riemann's explicit formula and calculations on the line $s=1/2$.  I don't know if there is any other argument for why zeros should exist with real part in $(0,1)$, except for analogy with other types of zeta functions (that did not exist in Riemann's time).