1. The problem is to understand the Riemann zeta function defined as $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$. 2. This is an infinite series where $s$ is a complex number with real part greater than 1 for convergence. 3. The formula sums the reciprocals of the natural numbers raised to the power $s$. 4. Important rules: - The series converges if $\text{Re}(s) > 1$. - For $s=2$, it converges to $\frac{\pi^2}{6}$. 5. Intermediate work involves evaluating partial sums or using analytic continuation for other $s$ values. 6. This function is fundamental in number theory and complex analysis, connecting to prime numbers and the distribution of zeros. Final answer: $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$