The following functions are called Riemannian functions: R(x)=0, if x = 0, 1 or an irrational number within (0, 1); R(x)= 1/q, if x=p/q(p/q is an approximate true fraction), that is, x is a rational number within (0, 1); This function is a special function, which was discovered by German mathematician Riemann and has a wide range of applications in advanced mathematics. In many cases, it can be used as a counterexample to verify some propositions to be proved in some functions. This function has an important application in calculus. 1September, 826 17, Riemann was born in Breselenz village in Hanover, northern Germany. His father is a poor country priest. He started school at the age of six, 14 entered the pre-university study, and 19 entered the University of G? ttingen, studying philosophy and theology according to his father's wishes, so as to follow his father's wishes and become a priest in the future. Because he loved mathematics since he was a child, Riemann listened to some math classes while studying philosophy and theology. At that time, the University of G? ttingen was one of the mathematical centers in the world, and some famous mathematicians such as Gauss, Weber and Steyr had taught in the school. Riemann was infected by the atmosphere of mathematics teaching and research here and decided to give up theology and specialize in mathematics. From 65438 to 0847, Riemann transferred to Berlin University and became a student of Jacoby, Dirichlet, Steiner and Eisenstein. 1849, he returned to the University of G? ttingen to study for a doctorate and became a student of Gauss in his later years. L85 1 year, Riemann received a doctorate in mathematics; 1854, he was hired as a part-time lecturer at the University of G? ttingen. 1857 promoted to associate professor; 1859, Dirichlet was hired as a professor to replace his death. Due to years of poverty and fatigue, Riemann began to suffer from pleurisy and tuberculosis less than a month after she got married in 1862, and spent most of the next four years in Italy for treatment and rest. 1866 died in Italy on July 20th at the age of 39. Riemann is one of the most original mathematicians in the history of world mathematics. Riemann's works are few, but extremely profound, full of creation and imagination of concepts. In his short life, Riemann has done a lot of basic and creative work in many fields of mathematics and made great contributions to world mathematics. The founder of complex variable function theory1The most unique creation of mathematics in the 9th century is the establishment of complex variable function theory, which is the continuation of people's research on complex number and complex variable function theory in the 8th century. Before 1850, Cauchy, jacoby, Gauss, Abel, Wilstrass and others had systematically studied the theory of single-valued analytic functions, but for multi-valued functions, only Cauchy and Pisser had some isolated conclusions. 185 1 year, under the guidance of gauss, riemann completed his doctoral thesis entitled "general theoretical basis of simple complex variable function", and later published four important articles in the journal of mathematics, further expounding the ideas in his doctoral thesis. On the one hand, he summarized the previous achievements about single-valued analytic functions, processed them with new tools, and established the theoretical basis of multi-valued analytic functions. Cauchy and riemann sum Wilstrass are recognized as the main founders of the theory of complex variable functions, and it was later proved that Riemann method is essential in dealing with the theory of complex variable functions. The thoughts of Cauchy and Riemann are integrated, and the thoughts of Wilstrass can be deduced from Cauchy-Riemann's point of view. In Riemann's treatment of multivalued functions, the most important thing is that he introduced the concept of "Riemann surface". Multi-valued functions are geometrically intuitive through Riemannian surfaces, and the multi-valued functions represented on Riemannian surfaces are single-valued. He introduced fulcrum, section line, defined connectivity on Riemannian surface, and studied the properties of functions, and obtained a series of results. The complex function handled by Riemann, single-valued function is an example of multi-valued function. He extended some known conclusions of single-valued functions to multi-valued functions, especially his method of classifying functions according to connectivity, which greatly promoted the initial development of topology. He studied Abel function, Abel integral and the inversion of Abel integral, and got the famous Riemann-Roche theorem. The first double rational transformation constitutes the main content of algebraic geometry developed in the late19th century. In order to perfect his doctoral thesis, Riemann gave several applications of his function theory in conformal mapping at the end of the paper, extended Gauss's conclusion about conformal mapping from plane to plane in 1825 to arbitrary Riemann surfaces, and gave the famous Riemann mapping theorem at the end of the paper. Riemann, the founder of Riemann geometry, made the most important contribution to mathematics. His research on high-dimensional abstract geometry and the methods and means to deal with geometric problems are a profound revolution in the history of geometry. He established a brand-new geometric system named after it, which had a great influence on the development of modern geometry and even the branches of mathematics and science. 1854, Riemann gave a speech to all the faculty and staff in order to obtain additional lecturer qualification at the University of G? ttingen. This speech was published two years after his death (1868), entitled "Hypothesis as the Basis of Geometry". In his speech, he briefly summarized all known geometries, including hyperbolic geometry, one of the newly born non-Euclidean geometries, and proposed a new geometric system, which was later called Riemannian geometry. In order to compete for the prize of Paris Academy of Sciences, Riemann wrote an article on heat conduction in 186 1, which was later called his "work in Paris". In this paper, his article 1854 is treated technically to further clarify his geometric thought. This article was included in his anthology 1876 after his death. Riemann mainly studies the local properties of geometric space, and he adopts differential geometry, which is opposite to the Euclidean geometry or non-Euclidean geometry of Gauss, Bolyai and Lobachevsky, which regards space as a whole. Riemann got rid of the shackles of Gauss and other predecessors who limited geometric objects to curves and surfaces in three-dimensional Euclidean space, and established a more general abstract geometric space from the perspective of dimensions. Riemann introduced the concepts of manifold and differential manifold, and called dimensional space manifold. A point in a dimensional manifold can be represented by a set of specific values of variable parameters, and all these points constitute the manifold itself. This variable parameter is called the coordinate of manifold and is differentiable. When the coordinates change continuously, the corresponding points traverse the manifold. Riemann takes the traditional differential geometry as the model, and defines the distance between two points on the manifold, the curves on the manifold and the included angle between the curves. Based on these concepts, the geometric properties of dimensional manifolds are studied. On the dimensional manifold, he also defined the curvature similar to Gaussian when studying the general surface. He proved that when his dimension on the dimensional manifold is equal to 3, the situation of Euclidean space is consistent with the results obtained by Gauss and others, so Riemann geometry is a generalization of traditional differential geometry. Riemann developed Gauss's geometric thought that the surface itself is a space, and studied the intrinsic properties of dimensional manifolds. Riemann's research led to the birth of another non-Euclidean geometry-elliptic geometry. In Riemann's view, there are three different geometries. The difference between them lies in the number of parallel lines made by a given point around a fixed straight line. If only one parallel line can be made, it is called Euclidean geometry; If you don't know any of them, it's elliptic geometry; If there is a set of parallel lines, we can get the third geometry, that is, Luo Barczewski geometry. Riemann therefore developed the space theory after Lobachevsky, ending the discussion about Euclid's parallel axiom for more than 1000 years. He asserted that objective space is a special kind of manifold, and foresaw the existence of manifolds with certain properties. These were gradually confirmed by later generations. Because Riemann considers the geometric space of any dimension, it has deeper practical value to the complex objective space. Therefore, in high-dimensional geometry, due to the complexity of multivariate differential, Riemann adopted some methods different from those of his predecessors, which made the expression more concise, and finally led to the birth of modern geometric tools such as tensor, external differential and connection. Einstein successfully used Riemann geometry as a tool to explain general relativity. Now, Riemannian geometry has become the necessary mathematical basis of modern theoretical physics. The Creative Contribution of Calculus Theory Riemann not only made pioneering work in geometry and complex variable function, but also went down in history with his outstanding contribution to the perfection of calculus theory which rose at the beginning of19th century. From the end of 18 to the beginning of 19 century, the mathematical community began to care about the imprecision of the concept and proof of calculus, the largest branch of mathematics. Porzano, Cauchy, Abel, Dirichlet and later Wells all devoted themselves to rigorous analysis. Riemann studied mathematics from Dirichlet in Berlin University, and had a deep understanding of Cauchy and Abel's work, so he had his unique views on calculus theory. 1854, Riemann needed to submit a paper reflecting his academic level in order to obtain additional lecturer qualification at the University of G? ttingen. What he handed in was an article about the possibility of expressing functions by trigonometric series. This is a masterpiece with rich contents and profound thoughts, which has far-reaching influence on perfecting analytical theory. Cauchy once proved that continuous functions must be integrable, and Riemann pointed out that integrable functions are not necessarily continuous. Cauchy and almost all mathematicians of his time believed in the relationship between continuity and differentiability. In the next 50 years, many textbooks "proved" that continuous functions must be differentiable. Riemann gave a famous counterexample of continuity and differentiability, and finally clarified the relationship between continuity and differentiability. Riemann established the concept of Riemann integral described in calculus textbooks, and gave the necessary and sufficient conditions for the existence of this integral. Riemann studied Fourier series in his own unique way, and extended Dirichlet condition, that is, Riemann condition on convergence of trigonometric series, and obtained a series of theorems on convergence and integrability of trigonometric series. He also proved that the terms of any conditionally convergent series can be rearranged appropriately so that the new series converges to any specified sum or divergence. Cross-century achievements of analytic number theory 65438+An important development of number theory in 2009 is the introduction of analytic methods and analytic results initiated by Dirichlet, while Riemann pioneered the study of number theory with complex analytic functions and achieved cross-century achievements. 1859, riemann published the paper "the number of prime numbers under a given size". This is an extremely in-depth paper, less than ten pages. He attributed the distribution of prime numbers to the problem of function, which is now called Riemann function. Riemann proved some important properties of functions, and simply asserted other properties without proof. In the more than one hundred years after Riemann's death, many of the best mathematicians in the world have been trying to prove his assertion, and in the process of these efforts, they have created new branches with rich contents for analysis. Now, except for one of his assertions, the rest have been solved as Riemann expected. That unsolved problem is now called "Riemann conjecture", that is, all zeros in the belt region are on a straight line (the eighth of Hilbert's 23 problems), which has not been proved so far. For some other fields, members of the Bourbaki school have proved the corresponding Riemann conjecture. The solution of many problems in number theory depends on the solution of this conjecture. Riemann's work is not only a contribution to analytic number theory, but also greatly enriches the content of complex variable function theory. Before the publication of Dr. Riemann's paper, the pioneers of combinatorial topology had made some scattered achievements in combinatorial topology, among which Euler euler theorem was the most famous one about the relationship between vertices, edges and faces of closed convex polyhedron. There are still some seemingly simple problems that have not been solved for a long time, such as the problem of the seven bridges in Konigsberg and the four-color problem, which prompted people to study combinatorial topology (then called position geometry or position analysis). However, the biggest motivation of topology research comes from Riemann's theory of complex variables. In the doctoral thesis of 185 1, and in the study of Abelian functions, Riemann emphasized some theorems that the study of functions necessarily requires position analysis. According to modern topological terminology, Riemann has actually classified closed surfaces according to genus. It is worth mentioning that he said in his dissertation that the idea that all functions are composed of connected closed regions (at spatial points) is the earliest functional idea. Betty, a professor of mathematics at the University of Pisa, once met Riemann in Italy. At that time, Riemann was ill and could not continue to develop her own ideas, so she taught her the method. Betty extended the topological classification of Riemannian surfaces to the connectivity of high-dimensional graphs, and made outstanding contributions in other fields of topology. Riemann is a well-deserved pioneer of combinatorial topology. The Open Source Contribution of Algebraic Geometry1In the second half of the 9th century, people became very interested in Riemann's method of studying the double rational transformation created by Abel integral and Abel function. At that time, they called the study of algebraic invariants and birational transformations algebraic geometry. In the paper of 1857, Riemann thinks that all equations (or surfaces) that can be transformed into each other belong to one class, and they have the same genus. Riemann calls the number of constants "quasi-modules", and the constants are invariant under the double rational transformation. The concept of "quasi-module" is a special case of "parametric module" at present, and studying the structure of parametric module is one of the hottest fields in modern times. Clebsch, a famous algebraic geometer, later became a professor of mathematics at the University of G? ttingen. He became more familiar with Riemann's work and made new progress in it. Although Riemann died young, it is universally acknowledged that the first step in the study of curve birational transformation was caused by Riemann's work. Rich achievements in mathematical physics, differential equations and other fields Riemann not only made epoch-making contributions to pure mathematics, but also paid close attention to physics and the relationship between mathematics and the physical world. He wrote some papers on heat, light, magnetism, gas theory, fluid mechanics and acoustics. He was the first person to deal with shock waves mathematically. He tried to unify gravity and light and study the mathematical structure of human ears. He summarized ordinary differential equations and partial differential equations abstracted from physical problems, and achieved a series of fruitful results. 1857, Riemann's paper "Supplement to Function Theory Represented by Gaussian Series" and an unpublished fragment written in the same year were included in their complete works. He dealt with hypergeometric differential equations and discussed the order linear differential equations of algebraic coefficients. This is an important document about the singularity theory of differential equations. /kloc-In the second half of the 9th century, many mathematicians spent a lot of energy on Riemann problems, but all failed. It was not until 1905 that Hilbert and Kellogg gave a complete solution for the first time with the help of the developed integral equation theory. Riemann has also made great achievements in the study of automorphic functions in the theory of ordinary differential equations. In his lecture on hypergeometric series of 1858 ~ 1859 and his posthumous work on minimal positive surfaces published by 1867, he established the automorphic function theory for studying second-order linear differential equations, which is now commonly known as Riemann-Schwartz theorem. In the theory and application of partial differential equations, Riemann creatively put forward a new method to solve the initial value problem of wave equations in the papers from 1858 to 1859, which simplified the difficulty of many physical problems. He also popularized Green's theorem; He has done outstanding work on Dirichlet's principle about the existence of solutions of differential equations ... Riemann's lectures on partial differential equations used in physics were later edited and published by Weber as "Differential Equations in Mathematical Physics", which is a historical masterpiece. However, Riemann's creative work was not unanimously recognized by the mathematics community at that time. On the one hand, his thoughts were too profound for people at that time to understand. Without the concept of free motion, Riemannian space with extraordinary curvature would be hard to accept, and it was not until the appearance of general relativity that the accusation was quelled. On the other hand, some of his work is not rigorous enough, such as abusing Dirichlet's principle when demonstrating Riemann mapping theorem and Riemann-Roche theorem, which has caused great controversy. Riemann's work directly influenced the development of mathematics in the second half of19th century. Many outstanding mathematicians have re-demonstrated the theorem asserted by Riemann, and many branches of mathematics have made brilliant achievements under the influence of Riemann's thought.