do all the non-trivial roots have real part 1/2?

Yes, all non-trivial roots of the Riemann zeta function are conjectured to have a real part equal to $ \frac{1}{2} $. This is known as the **Riemann Hypothesis**, one of the most famous and longstanding unsolved problems in mathematics.

### Understanding the Context:

- **Riemann Zeta Function**: The Riemann zeta function, $ \zeta(s) $, is a complex function defined for complex numbers $ s = \sigma + it $, where $ \sigma $ is the real part and $ t $ is the imaginary part.

- **Trivial Roots**: The trivial roots of $ \zeta(s) $ are the negative even integers: $ s = -2, -4, -6, \dots $.

- **Non-Trivial Roots**: These are the roots that are not at the negative even integers. According to the Riemann Hypothesis, all non-trivial zeros lie on the so-called "critical line" where $ \sigma = \frac{1}{2} $.

### Riemann Hypothesis:
The hypothesis asserts that if $ \zeta(s) = 0 $ and $ s $ is a non-trivial root, then $ s = \frac{1}{2} + it $ for some real number $ t $.

While this has been verified for many roots, the hypothesis remains unproven for all cases.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the significance of the Riemann Hypothesis in mathematics?
2. How does the Riemann zeta function relate to prime numbers?
3. What are the known results supporting the Riemann Hypothesis?
4. What would be the implications if the Riemann Hypothesis were proven true or false?
5. How are non-trivial zeros of the Riemann zeta function calculated?

**Tip:** The Riemann Hypothesis is part of the Clay Mathematics Institute's "Millennium Prize Problems," with a reward of $1 million for a correct proof or disproof.

Explore the concept of non-trivial roots in the context of the Riemann zeta function and the Riemann Hypothesis. Discover why all non-trivial roots are conjectured to have a real part of 1/2. Learn about the significance of the Riemann Hypothesis and its implications in mathematics.

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