In this paper, we will introduce a new algebraic system called the elliptic complex, and consider the distribution of zeros of the function $ L(s, \chi) $ in the corresponding complex plane. The key to this article is to discover the limiting case of the Generalized Riemann Hypothesis on elliptic complex fields and, taking a series of elliptic complex fields as variables, to study the ordinary properties of their distributions about the non-trivial zeros of $ L(s, \chi) $. It is on the basis of these considerations that we will draw the following conclusions. First, the zeros of the function $ L(s, \chi) $ on any two elliptic complex planes correspond one-to-one. Then, all non-trivial zeros of the $ L(s, \chi) $ function on each elliptic complex plane are distributed on the critical line $ \Re(s) = {1\over 2} $ due to the critical case of the Generalized Riemann Hypothesis. Ultimately we proved the Generalized Riemann Hypothesis.
Citation: Xian Hemingway. The Generalized Riemann Hypothesis on elliptic complex fields[J]. AIMS Mathematics, 2023, 8(11): 25772-25803. doi: 10.3934/math.20231315
In this paper, we will introduce a new algebraic system called the elliptic complex, and consider the distribution of zeros of the function $ L(s, \chi) $ in the corresponding complex plane. The key to this article is to discover the limiting case of the Generalized Riemann Hypothesis on elliptic complex fields and, taking a series of elliptic complex fields as variables, to study the ordinary properties of their distributions about the non-trivial zeros of $ L(s, \chi) $. It is on the basis of these considerations that we will draw the following conclusions. First, the zeros of the function $ L(s, \chi) $ on any two elliptic complex planes correspond one-to-one. Then, all non-trivial zeros of the $ L(s, \chi) $ function on each elliptic complex plane are distributed on the critical line $ \Re(s) = {1\over 2} $ due to the critical case of the Generalized Riemann Hypothesis. Ultimately we proved the Generalized Riemann Hypothesis.
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Xian Hemingway. The Generalized Riemann Hypothesis on elliptic complex fields[J]. AIMS Mathematics, 2023, 8(11): 25772-25803. doi: 10.3934/math.20231315
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