In this first part of the talk, we introduce the classical ideas that led to the study of the Riemann zeta function. We begin with Euler’s approach to evaluating series such as $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6},$$ and his discovery of the product representation over prime numbers. This naturally leads to the question of how far the function $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ can be extended beyond its initial domain $\Re(s) > 1$.
We then discuss Riemann’s analytic continuation of $\zeta(s)$ and the functional equation, which relate the values of the zeta function at $s$ and $1-s$. Using these tools, we describe the structure of the trivial zeros at the negative even integers and introduce the critical strip $$0 \le \Re(s) \le 1,$$ where all nontrivial zeros lie. Particular attention is given to the symmetry of zeros and the definition of the Riemann $\xi$-function, an entire function encoding the same information as $\zeta(s)$.
Finally, we review key historical results: von Mangoldt’s formula for the number of nontrivial zeros $$N(T) \sim \frac{T}{2\pi}\log\frac{T}{2\pi},$$ and Hardy’s proof that there are infinitely many zeros on the critical line. These ideas set the foundation for stating the Riemann Hypothesis and understanding why it lies at the center of modern analytic number theory.