The distribution of nontrivial zeros of the Riemann zeta function and the proof that they all lie on the critical line with a real part equal to 1/2 is a famous result known as the Riemann Hypothesis. Unfortunately, proving or disproving the Riemann Hypothesis remains an open problem in number theory, and it is one of the most difficult unsolved problems in mathematics.

The Riemann Hypothesis, proposed by the German mathematician Bernhard Riemann in 1859, states that all nontrivial zeros of the Riemann zeta function have a real part equal to 1/2. The Riemann zeta function is defined for complex numbers s with real part greater than 1 as the infinite sum ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + ..., and can be analytically continued to the whole complex plane apart from the point s = 1, where it has a simple pole.

Although the Riemann Hypothesis has been extensively tested and verified for the first 10^13 zeros, it has not been proven or disproven for all possible zeros. Many consequences of the Riemann Hypothesis have been established, and it is widely believed to be true due to the strength of the evidence supporting it. However, until a proof or counterexample is found, the distribution of nontrivial zeros on the critical line (with a real part equal to 1/2) remains unproven.

The Riemann Hypothesis is of great importance in number theory, as it has profound implications for the distribution of prime numbers. Many results in number theory rely on assuming the truth of the Riemann Hypothesis, and there are connections between the Riemann zeta function and other areas of mathematics, such as algebraic geometry and physics.

Efforts to prove the Riemann Hypothesis have led to significant advancements in the field of analytic number theory, and mathematicians continue to search for new techniques and insights that may eventually lead to a proof or disproof of this conjecture.

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