We define the Riemann zeta function as a function of a complex variable $s$ with the series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\operatorname{Re}s > 1 $ and then extend it analytically to ${\mathbb C} \setminus \{1\}$. We use a functional equation in which the gamma function plays an important role. The extended zeta function has a simple pole in 1 and so-called trivial zeros in $-2, -4, -6, \; \dots$. Later on, we express the Riemann zeta function as an infinite product called the Euler product and show that $\zeta$ has no zeros on the half-plane $\operatorname{Re} s \geq 1$. We use this fact in the proof of the prime number theorem which describes the asymptotic equivalence of the functions $\pi (x)$ and $x / \ln (x) $, where $\pi (x)$ denotes the number of primes less than or equal to a given positive real number $x$.
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