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Drakkith said:Best of luck to all those trying to solve it!
David Hilbert said:If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
The Extended Riemann Hypothesis is a conjecture in mathematics that extends the original Riemann Hypothesis, which is one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line with real part equal to 1/2.
The Extended Riemann Hypothesis was proposed by the mathematician Bernhard Riemann in 1859. He was trying to find a pattern in the distribution of prime numbers and came up with the Riemann zeta function, which is the key to the hypothesis.
Ramanujan's Sum is a mathematical function named after the Indian mathematician Srinivasa Ramanujan. It is defined as the sum of all positive integers that can be expressed as the sum of two cubes in two different ways. For example, 1729 is a Ramanujan's Sum because it can be expressed as 1^3 + 12^3 and 9^3 + 10^3.
The Extended Riemann Hypothesis and Ramanujan's Sum are connected through the Riemann zeta function. Ramanujan's Sum can be expressed as a special case of the zeta function, and the Extended Riemann Hypothesis predicts the behavior of the zeta function, including the distribution of Ramanujan's Sum. Therefore, solving the Extended Riemann Hypothesis would also provide insights into Ramanujan's Sum.
The Extended Riemann Hypothesis has significant implications in number theory and has connections to other areas of mathematics, such as algebra, analysis, and geometry. It also has practical applications in cryptography and prime number generation. Its proof or disproof would have a major impact on our understanding of the distribution of prime numbers and the nature of the Riemann zeta function.