1. The problem is to understand the function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, known as the Riemann zeta function. 2. This function is defined as an infinite series where $s$ is a complex number with real part greater than 1 for convergence. 3. The formula used is the infinite sum: $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ which means you add up terms $\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$. 4. Important rules: - The series converges when $\text{Re}(s) > 1$. - It can be analytically continued to other values of $s$ except $s=1$ where it has a simple pole. 5. Intermediate work involves evaluating partial sums for specific values of $s$ to approximate $\zeta(s)$. 6. For example, if $s=2$, then: $$\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = 1 + \frac{1}{4} + \frac{1}{9} + \cdots$$ which converges to $\frac{\pi^2}{6} \approx 1.645$. 7. This function is fundamental in number theory and complex analysis, especially in the distribution of prime numbers. Final answer: The Riemann zeta function $\zeta(s)$ is defined by the infinite sum $\sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\text{Re}(s) > 1$ and can be extended analytically elsewhere.