Introduction

Bornhard Riemann, a German mathematician and physicist, made important contributions to mathematical analysis and differential geometry, some of which contributed to the development of general relativity. paved the way.

His name appears in the Riemann zeta function, Riemann integral, Riemann geometry, Riemann lemma, Riemann manifold, Riemann mapping theorem, Riemann-Hilbert problem, Riemann Ideas in loop matrices and Riemannian surfaces.

He made his debut on the stage and gave a lecture entitled "On Assumptions as the Basis of Geometry", which created Riemannian geometry and provided the mathematical foundation for Einstein's general theory of relativity. He became a non-staff professor at the University of G?ttingen in 1857 and became a full professor after Dirichlet's death in 1859. Character experience

In 1826, he was born in Breselenz, a small town in the Kingdom of Hanover (now Germany). His father, Friedrich Bornhard Riemann, was a local Lutheran minister. He was the second of six children.

In 1840, Riemann moved to Hanover to live with his grandmother and entered middle school.

In 1842, after the death of his grandmother, he moved to Johanneum in Lüneburg.

In 1846, in accordance with his father's wishes, Riemann entered the University of G?ttingen to study philosophy and theology. During this period he attended a number of mathematics lectures, including Gauss's lectures on the method of least squares. With his father's permission, he switched to mathematics. During his college years, he went to the University of Berlin for two years and was influenced by C.G.J. Jacobi and P.G.L. Dirichlet.

In the spring of 1847, Riemann transferred to the University of Berlin and studied under Jacobi, Dirichlet and Steiner. Two years later he returned to G?ttingen.

In 1851, he received his doctorate. Riemann's signature

In 1851, he demonstrated the necessary and sufficient conditions for the differentiability of complex functions (ie the Cauchy-Riemann equation). The Riemann mapping theorem was elaborated with the help of Dirichlet's principle, which became the basis of the geometric theory of functions.

In 1853, the Riemann integral was defined and the criteria for the convergence of trigonometric series were studied.

In 1854, he carried forward Gauss's research on differential geometry of curved surfaces, proposed using the concept of manifold to understand the essence of space, and using the positive definite quadratic form determined by the square of the length of the differential arc to understand the metric, establishing Le The concept of Mann space included Euclidean geometry and non-Euclidean geometry into his system.

In 1854, he became a lecturer at the University of G?ttingen.

In 1857, he made his debut on the stage and gave a lecture entitled "On Assumptions as the Basis of Geometry", creating Riemannian geometry. And provided the mathematical basis for Einstein's general theory of relativity.

In 1857, he published a research paper on Abelian functions, which introduced the concept of Riemann surfaces, brought the theory of Abelian integrals and Abelian functions to a new turning point and conducted systematic research. Among them, Riemann surfaces were studied in depth from the perspectives of topology, analysis, and algebraic geometry. He created a series of concepts that had a profound impact on the development of algebraic topology, and clarified the Riemann-Loch theorem that was later supplemented by G. Rohe.

In 1857, he was promoted to non-staff professor at the University of G?ttingen. Riemann Atlas

In 1859, he succeeded Dirichlet as professor.

In 1862, he married Elise Koch.

On July 20, 1866, he died of tuberculosis in Selasca on his third trip to Italy to recuperate. Main results

In his paper on the distribution of prime numbers published in 1858, he studied the Riemann zeta function and gave the integral representation of the zeta function and the functional equation it satisfies. He proposed the famous The Riemann Hypothesis remains unsolved to this day.

In addition, he made significant contributions to partial differential equations and their application in physics. He even made important contributions to physics itself, such as thermal science, electromagnetic non-transdistance action and shock wave theory.

Riemann’s work directly affected the development of mathematics in the second half of the 19th century. Many outstanding mathematicians re-demonstrated the theorems asserted by Riemann. Under the influence of Riemann’s ideas, many branches of mathematics achieved brilliant achievements. . Riemann first proposed new ideas and new methods to study number theory using the function theory of complex variables, especially the ζ function, which created a new era of analytic number theory and had a profound impact on the development of function theory of single complex variables. He is one of the most original mathematicians in the history of world mathematics. Riemann did not write many works, but he was extremely profound and rich in the creation and imagination of concepts.

In November 2015, Nigerian professor Opeyemi Enoch successfully solved a 156-year-old mathematical problem, the Riemann Hypothesis, and received US$1 million (approximately RMB 6.3 million). Yuan) bonus. The Riemann Hypothesis was proposed by the German mathematician Bernard Riemann in 1859. It involves the distribution of prime numbers and is considered one of the most difficult mathematical problems in the world. In 2000, the Clay Mathematics Institute in the United States listed the Riemann Hypothesis as one of the seven millennium mathematical problems.

He has made important contributions to mathematical analysis and differential geometry. Differential equations also contributed significantly. Riemann

He introduced the theory of trigonometric series, thereby pointing out the direction of integral theory, laying the foundation for modern analytic number theory, and raising a series of questions; he initially introduced the concept of Riemann surfaces, which influenced modern topology Science has a great influence; in algebraic function theory, such as the Riemann-Noch theorem is also very important. In differential geometry, Riemannian geometry was established after Gauss.

His name appears in the Riemann zeta function, Riemann integral, Riemann lemma, Riemann manifold, Riemann space, Riemann mapping theorem, Riemann-Hilbert problem, Cauchy -Riemann equation, Riemannian idea in the loop matrix.