2024/04/09 by Donghao Liu, Liu, Dasheng
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2404.06306
openalex publication_date 2024/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this hypothesis. However, the problem has been unresolved for over a century. Here we show that at least one set of quadruplet-zeros exists outside the critical line through expanding the infinite product of the Riemann Xi zero function. We found that assuming there are no zeros outside the critical line will result in a contradiction with the known result that the reciprocal sum of all zeros of the xi-function is a constant, thereby refuting the Riemann Hypothesis. Furthermore, we give a lower bound estimation of a kind of summation formula for the zero points outside the critical line.