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A geologist like Riemann has almost anticipated the most essential features of the real world. -- Eddington

Georg Friedrich Bernhard Riemann (Georg Friedrich Bernhard Rie-mann) was born on September 17, 1826 in Bresrenz, a small village in Hanover, Germany. Riemann began to learn arithmetic at the age of about six, and his natural talent for mathematics showed immediately. At the age of 10, he studied advanced arithmetic and geometry from a full-time teacher named Schultz, and Schultz soon found himself following the student, who often had a better way to solve problems than he did.

Schmalfuss, the principal of Riemann High School, noticed Riemann's math talent, allowed him to enter and leave the library at will, and allowed him to skip math classes. On Schmalfuss's recommendation, Riemann borrowed Legendre's number theory. This was undoubtedly the beginning of Riemann's interest in the riddle of primes. Legendre has an empirical formula for estimating the approximate number of primes less than any given number. One of Riemann's most profound and enlightening papers belongs to this field. In fact, the Riemann conjecture, which emerged from his attempt to improve Legendre's formula, has become one of the most difficult mathematical problems today.

With regard to the number of primes less than a given number, the German version of Riemann conjecture appears in the famous paper "on the number of primes less than a given number". The problem discussed in this paper is to provide a formula to show how many primes are less than the known number n. In the attempt to solve this problem, Riemann had to study infinite series.

Where s is a complex number and makes the series convergent. With this constraint, the infinite series is a definite function of s, marked as

This is why the famous Riemann zeta function changes with s, and zeta (s) takes different values successively. What is the value of s and zeta (s) is zero? Riemann's guess is that for all s whose real part is 1max 2, that is,

This is the famous Riemann conjecture. Whoever proves that it is true or not will bring great honor to himself and will incidentally solve many extremely difficult problems in prime theory, other parts of advanced arithmetic, and some areas of analysis. In 1914, the English mathematician G H Hardy proved that infinitely many values of s satisfy this conjecture, but infinity is not all. Riemann conjecture is not the kind of problem that can be solved by elementary methods, and it is more difficult than Fermat's Great Theorem.

Riemann taught himself at an astonishing speed in high school, not only grasping the works of Legendre, a great mathematician, but also familiar with calculus and its branches by studying Euler's works. It is quite surprising that Riemann began from such an ancient starting point of analysis (Euler's method became obsolete by the mid-1840s due to the work of Gauss, Abel and Cauchy) and later became a successful analyst.

In 1846, at the age of 19, Riemann became a philosophy student at the University of Gottingen. But he couldn't let go of Stern's lectures on equation theory and definite integrals, Gauss on least square methods, and Goldsmith on geomagnetism. Riemann admitted all this to his father and asked for permission to switch to mathematics. Father agreed with all his heart.

After a year at the University of Gottingen, Riemann transferred to the University of Berlin and studied at Jacobi, Dirichley, Steiner and Eisenstein. He learned a lot from these masters-advanced mechanics and advanced algebra from Jacobi, number theory and analysis from Dirichlet, modern geometry from Steiner, and from Eisenstein, who was three years his senior, he learned not only elliptic functions, but also self-confidence, because he and the young master had fundamentally different views on how theory should develop. Eisenstein adhered to a wonderful formula, more or less modern Euler style; Riemann wanted to introduce complex variables, deriving the whole theory from a few simple general principles with the least calculation. The function theory of single and complex variables initiated by Riemann is very important in the history of modern science.

In 1849 Riemann returned to the University of Gottingen to complete his studies in mathematics and received his doctorate. People usually regard him as a pure mathematician. In fact, he has a wide range of interests. In fact, he spends as much time in physical science as he does in mathematics. If he could live 20 or 30 more years, he would probably become the Newton or Einstein of the 19th century. His ideas on physics were extremely bold in his time. It was not until Einstein completed his general theory of relativity that physicists realized that the physics predicted by Riemann was reasonable (Riemann was studied geometrically).

During his last three semesters at the University of Gottingen, Riemann attended lectures on philosophy and William Weber's course in experimental physics. The unfinished drafts of philosophy and psychology left after Riemann's death show that as a philosophical thinker, he is as original as he is in mathematics. At the same time, as a physical mathematician, Riemann is on the same level as Newton, Gauss and Einstein in intuition about things that are likely to have scientific application value in mathematics.

Riemann concluded in 1850 (at the age of 24)

It is possible to establish a complete, self-justifiable mathematical theory that deduces the processes seen in a real space full of matter (continuously filled space) from the basic laws of a single point, regardless of gravity, electricity, magnetism or static thermodynamics.

This may be explained by Riemann's abandonment of all the theories of "super-distance action" in physical science that are beneficial to field theory. In field theory, for example, the various physical properties of the "space" surrounding a "charged particle" are the object of mathematical study. Fascinated by his work in physics, Riemann put his pure mathematics aside for a while, and in 1850 he joined a mathematical physics class just opened by Weber, Ulrich, Stern and Listine.

In 1857, Riemann introduced the topological method into the function theory of simple complex variables. Gauss once predicted that topology would become one of the most important areas of mathematics, and Riemann realized this prediction in part through his invention in function theory.

Riemann made amazing progress by using his surfaces and their topological properties, especially in the field of Abelian functions. One problem in this respect is how to make a cut so that an n-leaf surface is equal to a plane. This high degree of spatial "intuition" is extremely valuable.

In early November 1851, Riemann submitted his doctoral thesis "the basis of the General Theory of functions of single complex variables" to Gauss for review. Riemann called in after Gauss had finished reading his paper, and Gauss told him that he had been planning for many years to write a special paper on the same topic. Gauss theory

The paper handed in by Mr. Riemann provides convincing evidence that the author has made a comprehensive and in-depth study of those parts of the problem discussed in this article, and that the author has a creative, active and real mathematical mind. with brilliant and rich creativity. The expression is clear and concise, even beautiful in some places. Most readers will want the arrangement to be clearer. The whole thesis is a valuable work with content, which not only meets the standards required by the doctoral thesis, but also far exceeds these standards.

Since 1853 (Riemann was 27), he has focused on mathematical physics. Because of his growing enthusiasm for physical science, his inaugural paper was delayed for a long time and was not completed until the end of the year. He has to give an inaugural speech before he takes up the position of lecturer. Gauss designated "the basis of geometry" as the theme of Riemann's speech, which Gauss had studied for 60 years, and he wanted to see how such a young man could deal with the problem. Riemann painstakingly prepared the speech and was very popular. Riemann's hypothesis as the basis of Geometry is not only a great masterpiece in mathematics, but also a masterpiece for recommendation.

Riemann's unique works on Abelian functions, classical works on hypergeometric series and differential equations for this series are very important in mathematical physics. In these two works, Riemann is unique in his own new direction. The generality and intuition of his method is unique to him.

Riemann's development of Abel function theory is different from that of Weierstras, just as moonlight is different from daylight. Weierstras's research is methodical and accurate in all the details. As for Riemann, he saw the whole, but ignored the details. Weierstras's method is arithmetic, and Riemann's method is geometric and intuitive. There is no point in saying that one is "better" than the other; neither method can be understood from a general point of view.

Overwork and lack of reasonable rest left Riemann with a nervous breakdown at the age of 31, and Riemann was forced to spend a few weeks in the mountain village of Hartz (where he met Dadkin). One evening, Riemann read Brewster's biography of Newton and found Newton's letter to Bentley, in which Newton himself asserted that medium-free teleaction was impossible. This pleased Riemann and inspired him to make an impromptu speech. Today, the "medium" that Riemann praises is not the glowing ether, but his own "curved space", or its reflection in relativistic space-time.

In 1858, Riemann wrote about electrodynamics. He wrote to his sister about the article

I have presented my findings on the close relationship between electricity and light to the Royal Society of Gottingen, and I heard that Gauss had conceived another theory about this close connection, which was different from mine, and told his close friends, but I am fully convinced that my theory is correct and will be recognized in a few years. As we know, Gauss soon withdrew his paper and did not publish it. Maybe he is not satisfied with it himself.

It seems that Riemann is too optimistic on this issue; Clark Maxwell's theory of electromagnetic fields dominates the field today. The current state of light and electromagnetic theory is too complex to be introduced here; it is sufficient to note that Riemann's theory has not been handed down.

Dirichley died on May 5, 1859, so Riemann became Gauss's second successor at the age of 33. During a visit to Berlin, he was dined by Borchett, Cuomer, Cronek and Verstras. Various societies, including the Royal Society of London and the French Academy of Sciences, awarded him honors as members. In short, he received the highest honor that a scientist usually gets. When he visited Paris in 1860, he became acquainted with first-rate French mathematicians, especially Hermit, and his praise of Riemann was almost endless. This year, 1860, is a memorable year in the history of mathematical physics, because in that year, Riemann began to focus on his paper "A problem on Heat conduction," in which he developed all the methods of quadratic differential forms. today the quadratic differential form is the basis of the theory of relativity.

Riemann's material life improved greatly with his appointment as a full professor, and he was able to get married at the age of 36. His wife, Eliza Koch, is a friend of his sister. Only a month after marriage, Riemann suffered from pleurisy in 1862 and had not yet fully recovered and suffered from lung disease. In Gottingen, he often said he wanted to talk to Dedkin about his unfinished work, but never felt strong enough to withstand a visit. He spent his last days in a villa in Seraska on Lake Maggiore. Riemann died on July 20, 1866, at the age of 39.

The greatness of Riemann as a mathematician is that the methods and new ideas he reveals for pure and applied mathematics are extremely universal and applicable to an infinite range.

On the basis of geometry, he regarded the whole of a huge problem as a coherent unity. Only one of his great works, the paper on the basis of geometry in 1854, can be introduced here. Riemann pointed out that because there are different lines and surfaces, there are different kinds of three-dimensional spaces; we can only rely on experience to find out which of these three-dimensional spaces we live in. In particular, the axiom of plane geometry holds within the limits of the test on the plane of a piece of paper, but we know that this piece of paper is actually covered with many small wrinkles, on which the axiom (the total curvature is not zero) is not true. Similarly, although the axioms of solid geometry are true for a limited part of our space within the limits of the experiment, we have no reason to think that they are also true for a very small part, he said; if it helps to explain physical phenomena, we may have reason to conclude that they are not true for a very small part of space.

'i 'm going to point out a way here so that these thoughts can be applied to the study of physical phenomena, 'Mr. Riemann said. I think in fact:

A small part of space actually has a property similar to a hill that is curved on a flat surface on average; the general laws of geometry do not hold there.

The quality of being curved or twisted, transitioning continuously from one part of space to another in the form of waves.

This change in spatial curvature really occurs in what we call (measurable or illusory) the movement of matter.

In the physical world, according to (perhaps) the law of continuity, nothing happens except this change.

I try to explain the law of the double inflection of this hypothesis in a general way, but I have not yet reached any definite results that can be announced.

Riemann also believes that his new geometry will prove to be scientifically important. As indicated at the end of his paper, ∶

Therefore, either the reality that forms the basis of space must form a discrete manifold, or we must find the basis of the measurement relationship outside it in the binding force acting on it.

The answer to these questions can only be obtained from the phenomenon that the idea has identified by experience (Newton assumes that this phenomenon is the basis), or from the successive changes required to do the facts in this idea that it cannot explain.

This leads us into another science, the field of physical science, which the object of this work does not allow us to enter today.

Riemann's work in 1854 gave geometry a new concept. The geometry he imagined was non-Euclidean geometry, but it was neither non-Euclidean geometry in the sense of Robachevsky and John Boyer, nor in the sense of Riemann's own obtuse-angle hypothesis, but non-Euclidean geometry in a broader sense dependent on the concept of measurement. It is a misunderstanding to regard measurement relation as the center of Riemann theory in isolation; this theory contains far more things than some operational measurement principle, and this is one of its main features. Any explanation of Riemann's concise paper can not explain all the connotations of this paper; however, we will try to explain some of his basic ideas, we will choose the concept of three-point ∶ manifold, the definition of distance, and the concept of curvature of manifolds.

A manifold is a kind of object, and the so-called object means that any member of this class can be completely determined by giving it a certain number assigned in a certain order to reflect the "countable" property of these member elements; and this design in a given order reflects the original characteristics of this "countable" property. Even though this statement may be even more difficult to understand than Riemann's definition, it is still a valid starting point for the beginning, which in general mathematics is equivalent to ∶ a manifold is an ordered "n-ary" array. The collection of xroomn) Two such n-tuple arrays. , xampn) and (yawn1) and (yellow1) The two n-ary arrays are equal if and only if the corresponding numbers in them are equal respectively.

If every such ordered n-ary array in a manifold happens to have n numbers, then the manifold is said to be n-dimensional. So we're back to talking about Cartesian coordinates. If (xonomi.xanthine 2, … Every number in) is a positive, zero, or negative integer, or if it is an element of any countable set, and if this is true for every n-ary array in the set, then the manifold is said to be discrete. If you count the number of X-ray, X-ray, … The manifold can be continuously valued (such as a point moving along a line), then the manifold is continuous.

This definition ignores the question of whether a collection of ∶ ordered n-ary arrays or something "represented" by these n-ary arrays is a "manifold". In this way, when we say that (xgraine y) is the coordinate of a point on the plane, instead of asking what "a point on the plane" is, we begin to use these ordered pairs of numbers (xmeny), where xpencil y independently takes all the real numbers. On the other hand, sometimes we focus our attention on what symbols such as (XMagol y) mean. In this way, if x is a person's age in seconds and y is his height in centimeters, we may be interested in that person rather than his coordinates, and the mathematics we explore is only concerned with coordinates. In the same way, geometry is no longer concerned with what "space" is. For a modern mathematician, space is just a manifold of the kind of numbers described above, and the concept of space is derived from Riemann's "manifold".

Riemann said of measurement, "Measurement consists of the superposition of quantities that need to be compared." Without this, it can only be compared when one quantity is part of another, and it can only be determined whether it is more or less, not how much it is. By the way, some kind of consistent and useful measurement theory is urgently needed in theoretical physics, especially in quantum mechanics and relativity.

Riemann once again descended from the general principles of philosophy to less mysterious mathematics and began to develop a definition of distance, which was extracted from his concept of measurement and has proved to be effective in both physics and mathematics.

Pythagoras's distance formula is

How to extend it to the surface? A straight line on a plane is equivalent to a geodesic on a surface; but on a sphere, for example, for a right triangle formed by a geodesic, the Pythagorean formula does not hold. Riemann extends the Pythagorean formula to an arbitrary manifold ∶ as follows.

Set

Is the coordinate of two points on a manifold, which are "infinitely close" to each other. For the sake of simplicity, we explain the meaning of nasty 4. The distance is:

The square root of. For a special choice of all g, a "space" is determined. So we can have

All other g is zero. The space considered in the theory of relativity has this general type, in which all g except gems 11, 22, 33 and 44 are zero.

In the case of n-dimensional space, the distance between adjacent points is defined in a similar way; the general expression contains the term 1max 2n (nasty 1). If we know the generalized Pythagorean formula for the distance between two adjacent points, finding out that the distance between any two points in space is a solvable problem in integrals. A space whose measurement (measurement system) is determined by the above-mentioned type of formula is called Riemannian space.

Curvature, as expressed by Riemann, is another generalization derived from ordinary experience. The curvature of a straight line is zero; the "measure" of the degree to which a curve deviates from the straight line may be the same at every point on the curve (as it is for a circle), or it may be different. at this point, the limit method must be applied to represent the "magnitude of curvature". Similarly, for a surface, its curvature can be measured by the degree of deviation from the plane, and the curvature of the plane is zero. This can be popularized and made more accurate as follows. For the sake of simplicity, we first explain the case of two-dimensional space, that is, what we usually imagine as a surface. By a formula that represents the square of the distance between adjacent points on a given surface

It can be known that the curvature of any point on the surface can be calculated by using the given function gforth 11 ~ (th), 12 ~ (th) and 22. It is meaningless to talk about the "curvature" of a space more than two-dimensional in ordinary language, but Riemann generalizes the curvature of Gauss and establishes an expression that includes everything g in the general case of n-dimensional space in the same mathematical way. it is mathematically the same type as Gauss's Gaussian expression for the curvature of a surface, and this generalized expression is what he calls the measure of spatial curvature. It is possible to show a visual representation of more than two-dimensional curved space, but it is probably as useless to intuition as giving a pair of broken crutches to a person without feet, because it is not helpful to understanding. and they're mathematically useless.

Riemann places an infinite number of "spaces" and "geometry" created for special purposes (for dynamics, or pure geometry, or physical science) within the competence of professional geometrists. It binds a large number of important geometric theorems into a tight bundle that can be easily dealt with as a whole. Riemann's achievements taught mathematicians not to trust any geometry or any space as a necessary model of human intuition.

Finally, Riemann's definition of curvature, his method for studying the design of quadratic differential forms, and his understanding of the fact that curvature is an invariant have found a physical explanation in the theory of relativity. It does not matter whether the theory of relativity has reached its final form; since the advent of the theory of relativity, our views on physical science have been different from those in the past. Without Riemann's work, this revolution in scientific thought would not have been possible unless someone later could create the concepts and mathematical methods created by Riemann.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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