The **Riemann Hypothesis** is one of the most famous unsolved problems in mathematics, first conjectured by Bernhard Riemann in 1859. It is related to the distribution of prime numbers and is deeply connected to number theory, particularly through the **Riemann zeta function**.

### Riemann Zeta Function
The Riemann zeta function is a complex function, defined for complex numbers $ s $ where $ \text{Re}(s) > 1 $, by the following series:
\[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
\]
It can be analytically continued to other values of $ s $, except at $ s = 1 $, where it has a simple pole.

### Riemann Hypothesis Statement
The hypothesis asserts that **all non-trivial zeros of the Riemann zeta function lie on the critical line**, which is the line $ \text{Re}(s) = \frac{1}{2} $ in the complex plane. More formally, the hypothesis claims that:

\[
\zeta(s) = 0 \quad \text{implies} \quad s = \frac{1}{2} + i t
\]
for some real number $ t $, where $ i $ is the imaginary unit.

### Importance of the Riemann Hypothesis
- **Prime Number Distribution**: The hypothesis is deeply linked to the distribution of prime numbers. If the hypothesis is true, it would give precise estimates for the number of primes less than a given number.
- **Analytic Number Theory**: Proving or disproving the hypothesis has far-reaching consequences in various areas of mathematics, including number theory, complex analysis, and mathematical physics.
- **Millennium Prize Problem**: The Riemann Hypothesis is one of the seven **Millennium Prize Problems** established by the Clay Mathematics Institute, offering a $1 million reward for a proof.

### Zeros of the Zeta Function
The zeros of the zeta function can be classified into:
1. **Trivial Zeros**: These occur at the negative even integers $ s = -2, -4, -6, \dots $.
2. **Non-Trivial Zeros**: These are complex numbers with real part $ 0 < \text{Re}(s) < 1 $, and the Riemann Hypothesis conjectures that their real part is exactly $ \frac{1}{2} $.

### Current Status
The Riemann Hypothesis remains unproven. However, many non-trivial zeros have been numerically verified to lie on the critical line, adding to the belief that the hypothesis might be true.

Would you like a deeper explanation on a specific aspect, or more examples of how this relates to prime numbers?

Here are 5 related questions you might explore:
1. How is the Riemann Hypothesis connected to the Prime Number Theorem?
2. What are some consequences if the Riemann Hypothesis is proven false?
3. How does the Riemann zeta function extend to values where $ \text{Re}(s) \leq 1 $?
4. What methods have been used to numerically verify zeros on the critical line?
5. How does the Riemann Hypothesis relate to other unsolved problems in mathematics?

**Tip**: Understanding the behavior of the zeta function near the critical line can give deeper insights into the distribution of primes.

Explore the Riemann Hypothesis, one of the greatest unsolved problems in mathematics, its relation to prime numbers, the Riemann zeta function, and the critical line where non-trivial zeros are believed to lie. Discover the implications of this conjecture on number theory and complex analysis.

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