1. The problem is to understand the Zeta function, specifically the Riemann Zeta function $\zeta(s)$, which is defined for complex numbers $s$ with real part greater than 1 by the infinite series: $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ 2. This function can be analytically continued to other values of $s$ except for $s=1$ where it has a simple pole. 3. Important properties include the Euler product formula: $$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$ which connects the Zeta function to prime numbers. 4. For real $s > 1$, the series converges absolutely and defines a smooth function. 5. The Zeta function satisfies the functional equation: $$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$$ which relates values at $s$ and $1-s$. 6. The zeros of $\zeta(s)$ in the critical strip $0 < \Re(s) < 1$ are of great interest in number theory (Riemann Hypothesis). 7. To evaluate $\zeta(s)$ numerically for $s > 1$, sum the series up to a large $N$: $$\zeta(s) \approx \sum_{n=1}^N \frac{1}{n^s}$$ with error decreasing as $N$ increases. This explanation introduces the Zeta function, its definition, key properties, and significance in mathematics.