The **Riemann Hypothesis** is one of the most famous unsolved problems in mathematics. It concerns the distribution of prime numbers and is closely related to the Riemann zeta function, a complex function denoted as $\zeta(s)$, defined for complex numbers $s = \sigma + it$, where $\sigma$ and $t$ are real numbers.

The hypothesis, proposed by the German mathematician **Bernhard Riemann** in 1859, asserts that:

**All non-trivial zeros of the Riemann zeta function have a real part of $ \frac{1}{2} $.**

### Breaking it down:
1. **The Riemann Zeta Function:**
   - For real numbers $s > 1$, the zeta function is defined as:
     \[
     \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
     \]
   - This function can be extended to complex numbers and has both "trivial" and "non-trivial" zeros.

2. **Trivial Zeros:**
   - The trivial zeros of $\zeta(s)$ occur at the negative even integers: $s = -2, -4, -6, \dots$.

3. **Non-trivial Zeros:**
   - The non-trivial zeros are complex numbers where $\zeta(s) = 0$. Riemann conjectured that the real part of these zeros is always $ \frac{1}{2} $. That is, if $ s = \sigma + it $ is a non-trivial zero, then $ \sigma = \frac{1}{2} $.

### Why It Matters:
- The Riemann Hypothesis is crucial because it provides deep insights into the **distribution of prime numbers**. Specifically, it connects the spacing and occurrence of primes to the behavior of the zeta function.
- Verifying the hypothesis would improve our understanding of how primes are distributed, particularly in large numbers.

### Current Status:
- The Riemann Hypothesis is still unproven. It is one of the **Millennium Prize Problems**, for which the Clay Mathematics Institute offers a $1 million prize for a correct proof or disproof.

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**Relative Questions:**
1. How is the Riemann zeta function related to prime numbers?
2. What are the applications of the Riemann Hypothesis in number theory?
3. Can you explain the significance of the zeros of the zeta function?
4. What are some major attempts made to prove the Riemann Hypothesis?
5. What is the relationship between the Riemann Hypothesis and prime number theorems?

**Tip:** Understanding complex analysis can greatly help in studying the Riemann zeta function and the hypothesis itself.

Explore the Riemann Hypothesis, a crucial unsolved problem in mathematics. Learn how it connects the distribution of prime numbers to the zeros of the Riemann zeta function and delve into its profound implications in number theory. Suitable for university-level students.

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