The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics. It asserts that the real part of every non-trivial zero of the Riemann zeta function [tex]\(\zeta(z)\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
To understand the details, let's delve deeper into the elements involved:
1. Riemann Zeta Function:
[tex]\[
\zeta(z) = \sum_{n=1}^{\infty} \frac{1}{n^z}
\][/tex]
for [tex]\(\text{Re}(z) > 1\)[/tex]. It can be analytically continued to other parts of the complex plane except for [tex]\(z = 1\)[/tex].
2. Non-Trivial Zeros:
These are the zeros of [tex]\(\zeta(z)\)[/tex] that lie in the "critical strip" where [tex]\(0 < \text{Re}(z) < 1\)[/tex]. The Riemann Hypothesis posits that these non-trivial zeros all have a real part equal to [tex]\(\frac{1}{2}\)[/tex].
Unfortunately, proving this rigorously is far beyond the scope of basic mathematical techniques and has not been accomplished. However, extensive computational evidence and various mathematical properties suggest this pattern holds true.
3. Key Steps:
- Euler Product Formula:
[tex]\[
\zeta(z) = \prod_{p \text{ prime}} \left( 1 - \frac{1}{p^z} \right)^{-1} \quad \text{for} \ \text{Re}(z) > 1
\][/tex]
This connects the zeta function to prime numbers.
- Analytic Continuation and Functional Equation:
The zeta function can be extended to a meromorphic function on the whole complex plane and satisfies a certain symmetric functional equation:
[tex]\[
\zeta(z) = 2^{z} \pi^{z-1} \sin \left( \frac{\pi z}{2} \right) \Gamma(1 - z) \zeta(1 - z).
\][/tex]
This equation inherently links the values of [tex]\(\zeta(z)\)[/tex] and [tex]\(\zeta(1-z)\)[/tex], central to studying the zeros.
4. Symmetry and Critical Line:
From the functional equation, it's known that if [tex]\(z\)[/tex] is a zero of [tex]\(\zeta(z)\)[/tex], then [tex]\(1 - z\)[/tex] is also a zero. Since zeros appear symmetrically with respect to the critical line [tex]\(\text{Re}(z) = \frac{1}{2}\)[/tex], the hypothesis naturally emerges: the real part of every zero should be [tex]\(\frac{1}{2}\)[/tex].
In conclusion, while this problem has profound consequences and connections to number theory and complex analysis, it remains unproven. Advanced mathematical tools such as complex analysis, computational techniques, and deep insights into analytic number theory are pivotal towards eventually resolving the Riemann Hypothesis.
Note: The answer containing numerical values, `(20, 29)` related to the growth in the number of computers, is part of a different context, possibly another problem. Hence, it doesn't apply here to the discussion of the real part of non-trivial zeros of the Riemann zeta function.
