The equation \( \zeta(s) = 0 \) refers to the Riemann zeta function, denoted by \( \zeta(s) \), where \( s \) is a complex number. The equation is significant because it describes the zeros of the Riemann zeta function, which are central to the **Riemann Hypothesis**.

The **Riemann zeta function** is defined as:

\[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
\]

for \( s \in \mathbb{C} \) with \( \Re(s) > 1 \). It can be analytically continued to other values of \( s \) (except \( s = 1 \)) using more complex formulas.

### Zeros of the Riemann zeta function:
1. **Trivial zeros**: The function has trivial zeros at negative even integers: \( s = -2, -4, -6, \dots \).
2. **Non-trivial zeros**: The famous non-trivial zeros are conjectured to lie on the so-called "critical line" where \( \Re(s) = \frac{1}{2} \). This is the essence of the Riemann Hypothesis, which states that all non-trivial zeros have a real part of \( \frac{1}{2} \).

If you want to dive deeper, feel free to ask!

### Questions for further exploration:
1. What is the significance of the non-trivial zeros in the Riemann Hypothesis?
2. How does the Riemann zeta function relate to prime numbers?
3. What are the methods for computing the zeros of \( \zeta(s) \)?
4. Can you explain the analytic continuation of the zeta function?
5. What role do the trivial zeros play in number theory?

**Tip**: Understanding complex analysis, particularly concepts like analytic continuation and the behavior of functions in the complex plane, is crucial for grasping the Riemann zeta function fully.

Explore the equation ζ(s) = 0, focusing on the Riemann zeta function and its critical zeros. Learn about the Riemann Hypothesis, the trivial and non-trivial zeros, and the analytic continuation of the zeta function. Suitable for advanced undergraduate or graduate-level students.

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