Number theory provides us with an inexhaustible supply of interesting truths – truths that are not isolated but intrinsically linked, and as our knowledge grows, we discover new, and sometimes completely unexpected, connections between them. ”
Gauss. Written by |Ding Jiu (Professor, Department of Mathematics, University of Southern Mississippi, USA).
Reader, roll a piece of paper into a cylindrical shape, find a pencil tip, and place the bottom of it against the outside of the cylinder, with the tip facing outward perpendicular to the cylindrical surface. If you keep the two perpencils, make a circle around the cylindrical plane, or more generally, make the pencil perpendicular to the cylindrical plane, and move it around any closed curve on it that does not cross the circumference of the boundary, and you will see that the point of the pencil moves continuously, and finally returns to its original position. If you put the bottom of the pencil tip against the inside of the paper cylinder and do the same thing in a circle, the result will be the same. This means that the cylindrical face is "two-sided", and it has an inner and outer side. By specifying a fixed side on one of its two sides, the "right-hand rule" is relied upon to determine the orientation of any closed curve on the surface—forward and backward. This is a geometric phenomenon that every child can understand.
Readers who have studied surface integrals know that surfaces that are integral regions must be side-definite, otherwise surface integrals cannot be discussed. In the 80s of the last century, Professor Li Tianyan, my Ph.D. supervisor in the Department of Mathematics at Michigan State University, told me that he taught his junior high school son the concept of topology: take a narrow piece of paper, instead of gluing two short opposite sides together to form a short cylindrical surface like the above; Instead, one of the short sides is twisted 180 degrees and then adhered to the other short side. This also results in a paper surface. He then asked his son to do the same test as in the previous section, and it turned out that when the pencil went around a closed circuit in one direction that was almost the same as the long opposite side, and kept it perpendicular to the surface, the pencil tip terminated in the opposite direction to the original direction! Of course, this phenomenon does not occur when the closed circuit is small enough to be a circle around a point on the surface, but the existence of the closed circuit that leads to the "reversal of direction" anomaly fully demonstrates that this strange surface has a topological property that is completely different from that of ordinary cylindrical surfaces.
This strange surface is "one-sided" and is not approved by the room guards in the Calculus Building, but it is not only visual, but also rich in connotation, and its professional name is "M Bius Strip", after the discoverer.
1. The surname of the German mathematician and astronomer August Ferdinand M Bius (1790-1868). Another discoverer a few months before him was the German mathematician Johann Benedict Listing (1808-1882). The Möbius strip is the most well-known mathematical discovery in Möbius's life, because people understand it at a glance. However, the Möbius inversion formula, which was derived from his lesser-known mathematical work, is the subject of this article.
Mobius inversion.
The most primitive idea of the Mobius inversion formula is that we are familiar with the series part and the simple bilateral relationship between the series and the series of common terms. The sum of the parts of the first n terms of the series is. Conversely, the nth term of the series can be written as a=s-s (convention s=0). If you define a special series: =1, =-1, and when n 2, =0. Then the above part and the "mutual expression" with the general term are if and only if.
There are many generalizations and variations of the Möbius inversion formula, but the most famous and simple one is a "classic" with many uses in number theory and combinatorics. In order to understand this original formula, several elementary terms need to be introduced. First of all, the so-called "inversion" is a generalization of the concept of inverse functions in algebra in middle school. When the function y=f(x) reflects different independent variable values x into different function values on the defined domain, the function f leads to the corresponding inverse function f, which reflects the function value y of f back to the value of the independent variable that caused the value: x=f(y). In this way, all x in the function definition domain and all y in the value range establish a pair of "inversion relations": y=f(x) if and only if x=f(y). As an example, the inverse of the function y=x is x=y. Although a function has an inverse function, we can't write an algebraic expression for it. The inverse representation of invertible and inverse functions in elementary mathematics can be generalized to higher and more abstract mathematics.
For example, if all the sequences defined on the whole natural numbers are written as x, and if there is a correspondence t that mirrors each sequence of x into a sequence, this transformation is generally referred to as an "operator", especially in the discipline of functional analysis. If t mirrors a different sequence to a different sequence, then there is the inverse operator t, which mirrors each sequence in the range of t back to its ** sequence, i.e., =t if and only if =t. This also gives an inversion relation.
Now, we can describe the classic Möbius inversion formula in number theory. Let f be an "arithmetic function", i.e., its domain is the set of all natural numbers, and the value of the function is a complex number. Naturally, an arithmetic function can be equated with a sequence of its values on all natural numbers. In number theory, if the natural number d is a factor of the natural number n, i.e., n=dq, where q is also a natural number, then this relation is written as d|n。For all natural numbers n, the following formula.
A new arithmetic function g is defined. Then the operator t that mirrors f to g has an inverse operator, and its expression is the so-called Möbius inversion formula.
i) where is defined above, it is called the "Möbius function": 1)=1;If n is the product of k heterogeneous primes, then (n)=(-1); If the prime decomposition of n contains a prime square, then (n)=0.
Historically, the source of the Möbius inversion formula was an article published in 1832 by Professor Möbius of the University of Leipzig in German**, which translates to English as "A Special Type of Series Inversion"). However, the "inversion problem" studied in this paper has nothing to do with (*) and (i) above, but seeks the relationship between the coefficient series and the original coefficient series in the inversion series transformation f(x)= of the series of series transformations. However, in this article that left a name on the history of mathematics, he gave the expression of the Möbius function named by later generations, as well as its factors and formulas.
Möbius function.
Given the key role that the Möbius function plays in inverting formulas, let's take a look at its fundamental properties. Let's familiarize ourselves with the first dozen numbers in the Möbius sequence: (1)=1, (2)=-1, 3)=-1, (4)=0, 5)=-1, 6)=1, 7)=-1, 8)=0, 9)=0, 10)=1, 11)=-1, 12)=0. The first fundamental property of the function is that it is multiplicative, i.e., as long as there are two natural numbers m and n that are mutually prime (there is no positive common factor except 1), the equation (mn) = (m) (n) holds. In fact, when mn=1, m=n=1, so (mn)=1= (m) (n). If mn>1, set m=p....p and n = q....q, where p ,..., p, and q,..., q is a different prime number, then (mn)=(-1)=(-1)(-1)= (m) (n). The above equation also holds when m=1 or n=1 (note that 1 is not a prime number). Now at least one of m and n is a factor with the square of the prime number, then the square of the prime number is also a factor of mn, so (mn)=0= (m) (n). The above is a direct proof, and as an exercise, the reader can also give a second proof using mathematical induction, which is a good opportunity to train the brain. From 0 = (4)≠ (1)(-1) = (2) (2), the Möbius function is not "completely multiplicative", i.e., the equation (mn) = (m) (n) does not always hold.
By definition (1)=1. Let's prove a very useful equation: for any natural number n,greater than 1
For example, when n=20=2·2·5, its positive factors are 1, 2, 4, 5, 10, 20, hence there.
The above example hides the idea of proof of equation (1). According to the fundamental theorem of arithmetic, let the prime number of n be decomposed, and since the factor d of n is a multiple of the square number (d)=0, the sum on the left side of equation (1) only needs to be considered as p and p ,..., the product of some dissimilar numbers in p, and d = 1. These d are: 1=c(k, 0) 1, c(k, 1) p, c(k, 2) pp, ....c(k, k) = 1 pp....p, where c(k, i)=k!/[i!(k-i)!] is the number of all groups that select i at a time from k objects to form a group. Hence the binomial theorem,
Now let's set out to prove the Möbius inversion equation (i). First, according to the definition of the arithmetic function g, the order of the sum is exchanged (the principle is equivalent to dividing a finite set of numbers into groups in two ways, adding the numbers in the group and then adding the sum of the sum numbers in each way. In the simplest case, a set of numbers arranged in a long matrix is added, whether they are added line by row or column by column, and the result is the same, i.e., the right end of (i).
When c = n, and when 1 c
This proves (i).
From the arithmetic function f to the function value g(n) of the arithmetic function g, since the definition and inversion of equation (i) are only expressed in the form of finite sums, we only use the factors of the Möbius function and equation (1) to prove the Möbius inversion formula (i) in the "elementary place". In the same way, it can be shown that if f and g satisfy (i), then they also satisfy (*) The M Bius transform of F and the inverse M Bius transform of F are called f. Note that there is also an English mathematical term m bius transformation that is also translated into "Möbius transformation" in Chinese, which refers to a linear fractional transformation that mirrors complex numbers to complex numbers w=(az+b) (cz+d).
If f and g are replaced by in f and in g respectively in the Möbius transform, then (*) and (i) imply the following Möbius inversion formula in the form of multiplication.
If and only if
mi) The Möbius function is productive, and its factor summation arithmetic function is usually denoted as , and (n)=satisfying (1)=1 and (n)=0(n>1). Obviously, it is also a product function. This property can be generalized to the general conclusion that if the arithmetic function f is productive, then the arithmetic function g defined by (*) is also productive. It can be proved in this way: let the natural number m and n mutual. Defined by (*).
Because m and n do not have a positive common factor other than 1, d = ab, where a|m and b|n。Obviously, A and B are mutual, so there is.
Dirichlet convolution.
Readers who have studied Fourier transforms will be familiar with convolution operations between functions. The convolution f*g of the two functions f and g is defined as the integral of the product of one function and the other function after reflection and displacement, representing how the shape of one function is changed by the other. If the defined domains of f and g are both the entire real number axis, then their convolutions are. Using the variable substitution method of the integral, it is easy to prove that f*g=g*f, that is, the convolution operation is full of ** change law. The convolution theorem in Fourier analysis says that if f and g are Fourier transforms of f and g, respectively, then the inverse Fourier transform of the product of f and g is the convolution of f and g. There is a similar convolution theorem for the Laplace transform, which is commonly used in engineering mathematics.
So, is the idea and method of convolution also related to the "Möbius inversion"? Of course! This is the Dirichlet convolution used in number theory for arithmetic functions, and this concept is simply a direct extension of Möbius inversion. Its definition is very similar to the expression at the right end of Möbius inversion equation (i), except that the Möbius function is replaced by a general function: let f and g be arithmetic, then the Dirichlet convolution of f and g is an arithmetic function.
It is evident that Dirichlet convolutions are also commutative like integer multiplication. In addition, under the addition of functions and the "multiplication" connotation of Dirichlet convolution, the totality of all arithmetic functions also forms an exchangeable ring like the whole of all integers, called Dirichlet rings. The multiplicative unit of an integer ring is a positive integer 1, while the multiplicative unit of a Dirichlet ring is the arithmetic function mentioned earlier, whose official name is "identity arithmetic function" and has an identity. Naturally, it is not a coincidence. In fact, using the same method as the Möbius inversion formula above, it can be quickly verified that f* = *f=f.
In addition, Dirichlet convolutions, like integer multiplication, satisfy the associative and distributive properties: (f*g)*h=f*(g*h) and f*(g+h)=f*g+f*h. In the case of Dirichlet rings, if and only if the arithmetic function f satisfies f(1)≠0, it has the Dirichlet inverse, i.e., there is an arithmetic function f such that f*f= . In particular, the Dirichlet inverse of the constant function 1 is the Möbius function, i.e., there is the relation 1* = required in the next argument. Here we have used 1 to represent a function with a value of 1 everywhere on a set of natural numbers, and its name is "unit arithmetic function".
With the powerful tools of convolution, we can give a more concise proof of the Möbius inversion formula. First of all, the equation (*) can be written as a convolutional form g=f*1. Further, the derivation process from right to left of inversion equation (i) is.
Conversely, under the condition that (i) is true, it is deduced that (*) is true in the following steps:
It can be seen that in the context of Dirichlet convolution, the formulation of the classical Möbius transformation is:
g=f*1 if and only if f=g*.
The average science and engineering student probably learned about the name of the German mathematician Gust** Lejeune Dirichlet (1805-1859) from the Fourier series or partial differential equation boundary value problems, but do not mistake him for only "analytical mathematics", as almost all mathematicians today are proficient in one craft. He is also a master of number theory and pioneered the branch of analytic number theory. The modern definition of function is also derived from him, allowing secondary school students around the world today to benefit from this most plausible definition.
Since the Möbius inversion is only a "synonym" for the fact that the Dirichlet inverse of the unit arithmetic function 1 is the Möbius function, the original Möbius transform double formulas (*) and (i) can be immediately generalized to the following general inversion formula: Assuming that the arithmetic function has a Dirichlet inverse, then.
If and only ifThe more concise and striking convolution form is g= *f if and only if f= *g. If you think of the convolution sign as an arithmetic multiplication sign that elementary school students know, it would be as simple as 6=2 3 if and only if 3=2-1 6. It can be seen that abstract mathematics is not so difficult to understand.
We give another generalization of the classical Möbius inversion formula, which generalizes the arithmetic function defined on the set of natural numbers to the complex value function defined in the domain [1, . To show the difference from the previous defined domain, the function here will be represented by a capital letter. Let f and g be two functions that reflect [1, into a set of complex numbers, satisfying the equation.
where [x] is the maximum natural number less than or equal to x. We will deduce the following inversion formula.
In fact, just use the same method as proof (i) to deduce from the right end of ( ) to the left end:
The second equal sign above is because of the grouping by mn=k, rearranging the order of summation.
The generalized forms corresponding to the general formulas ( ), ( ) and ( ) in the discrete case are:
If and only ifEuler's function. Since the classical Möbius inversion formula was created for number theory, it seems unreasonable not to give a specific application of it in number theory. Let's take the well-known Euler function in number theory as an example. This function was introduced by Leonhard Euler (1707-1783) in 1763, and its value (n) at the natural number n is defined as the number of natural numbers that are not greater than n and are coprime with n. The first ten Euler function values are (1)=1, (2)=1, (3)=2, (4)=2, (5)=4, 6)=2, 7)=6, 8)=4, 9)=6, 10)=4.
Since every natural number is the product of prime numbers, we calculate how many (p) equals, where p is the prime number. The greatest common factor between the natural numbers of p and p from 1 to p can only be 1, p, p,..., p, so that the natural numbers with which the greatest common factor with it is greater than 1 are p, 2p, 3p,..., pp=p, they have a total of p, and the remaining natural numbers are mutualized with p, so there is the formula (p)=p-p.
The Euler function is integrative, i.e., for any of the coprime natural numbers m and n there is (mn)= (m) (n). We only briefly prove that m = p and n = q, where p and q are heteroprimes, and the same method can be proved in general. From the previous paragraph, we know that in the natural numbers less than p, there are (p)=p-p and p mutisoprimes, and the set of them is p; Similarly, in a natural number less than q, there is (q)=q-q and q are mutually prime, and their set is q. According to the Chinese remainder theorem, the product p q of these two sets has a one-to-one correspondence with all those natural numbers that are not greater than pq and are interprime with them. In other words, let the number A in p and the number b in q be given and the number ab
For the prime factorization of any natural number, the product of Euler's function is determined, which is known as Euler's product formula.
The Chinese remainder theorem used above is also known as Sun Tzu's theorem, and this "Sun Tzu" has nothing to do with "Sun Tzu's Art of War". In the "Sun Tzu Sutra" during the Northern and Southern Dynasties, there is an arithmetic problem: "There are things that do not know their number, three or three are left, five or five are left, and seven or seven are left." Ask about the geometry of things? The answer to the minimum number is 23, and the theorization of its solution becomes the "Sun Tzu's theorem" about a system of unary linear congruence equations. A special version of only two equations is given here: let the integers m and n be mutually primed. then for any integers a and b, a congruence equation.
There is a solution x = adn + bcm, where the integers c and d satisfy cm + dn = 1.
Now let's sum the factors of Euler's function: . How much is it? When n=p, then.
When n = pq, then each factor of n can be written as the product of the factor of p and the factor of q, so there is.
In the case of the right general case, it turns out that it is essentially the same. The formula that was demonstrated.
Founded by Carl Friedrich Gauss (1777-1855). If we use the Dirichlet convolutional form, it is *1=id, where id is the identity function.
The Mobius inversion formula is now available. Applying (i) to the Gaussian equation above gives the Euler function explicit expressed in the Möbius function.
The above expression provides a clue to finding the Dirichlet inverse of Euler's function. Removing n from the (n) expression and moving d up to the denominator into a molecule defines an arithmetic function. And then, therefore. Circular polynomial.
Let's use the Möbius inversion formula for a class of polynomials. The complex solution of the polynomial equation z-1=0 is called the nth root of 1, and they are.
, then the above n roots can be written as the power of : , where the roots of the power k and n are primitive roots of 1. An equivalent definition of the primordial root is that it is the root of z-1, but not the root of any lower polynomial z-1. The original root has the property: implicit n|l。
Given n, the first polynomial whose root happens to be all primordial roots is called a circular polynomial of order n, i.e.
In the above equation, gcd(k, n) represents the greatest common factor of k and n.
For any nth root z, the smallest natural number d such that the equation z=1 holds is called the order of z, which satisfies d|n。This property can be used to prove the following polynomial equation.
In fact, let z be a zero point of the right polynomial in (**, then for a positive factor d of n, there is z=1. Let n=dm, then z=(z)=1=1. On the other hand, if z = 1, then the order d of z is divisible by n, so z is the primitive d root of 1, i.e. it is a zero point of the circular polynomial. (*Formula.)
Apply the multiplicative form of the Möbius inversion formula (MI) to the explicit expression of the polynomial of the score circle:
Infinite series. The series discussed so far are all "infinite series", that is, sums of infinite numbers. Considering several infinite series below, when performing the Möbius inversion surgery of "series general term group rearrangement", it is necessary to ensure that the operation is correct, and a sufficient condition for the success of the operation is the "absolute convergence" of the relevant series, and this assumption will be given without explanation once the infinite series is released. The reason for this is simple: merely a conditionally converged series can rearrange the series of general terms so that the new series changes its sum. Let's consider first a special class of series named after the surname of the polymath Johann Heinrich Lambert (1728-1777). For infinite sequences, it is assumed that |x|<1, using the formula for summing proportional series, there is.
The left end of the above equation is called the Lambert series, and the right end states that it is equal to the power series, where the sum satisfies (*) in particular, if, due to, there is an identity.
If you take, then. Replace x=e with a variable in the Lambert series formula to get another form of it:
A similar approach can be applied to the so-called Dirichlet series. Multiply the series expression of the Riemann function by the Dirichlet series, using the same technique as Lambert series.
In particular, f(n) is the Möbius function (n) because (see the previous Dirichlet convolution equation 1* = ), we get the series expression of the reciprocal of the function.
A more general equation can be deduced by the proof idea of the ($ formula.
Unexpected connections.
At this point, none of the Möbius inversion formulas and their applications have gone beyond the realm of pure mathematics. Doesn't it find application in other disciplines? At least the British number theorist Godfrey Harold Hard (1877-1947) firmly believed that only "low-grade mathematics" such as calculus could be played with by applied scientists, and number theory was regarded as the "queen of mathematics" by Gauss, the prince of mathematics, who could only appreciate its beauty but could not be assigned tasks. The global physics community doesn't seem to have really paid much attention to the Möbius inversion formula. Until 1990, the top physics journal "Physical Review Letters" (abbreviated as PRL) published a ** signed by Chinese alone in Volume 64, No. 11, which even alarmed the then editor-in-chief of "Nature" and published a full-page review for it.
The Chinese scholar, Chen Nanxian (1937-), graduated from the Department of Physics at Peking University and received his Ph.D. in electrical engineering and science from the University of Pennsylvania in 1984. He was elected an academician of the Chinese Academy of Sciences seven years after the publication of this unique article. The title of the article is "Modifying the Möbius Inversion Formula and Its Application in Physics rev. lett. 64，1193；1990）”。
So, what is Professor Chen's modified Möbius inversion formula? The biggest difference between it and the classical formula in form is that the new formula is a pair of infinite series expressions, and the latter is an inverse representation of the former (for the sake of consistency with the notation in this article, I changed a to g, b to f, and x in the original text).
If and only if (cnx)
At first glance, the above formula differs from the previous formulas (** and ( ) only in that the infinite series are replaced by infinite series. It is precisely the leap from poverty to infinity that makes people feel that the proof will not be so "primary", and the limit thinking must be inserted to help. The fact is that the proof provided by the author in the appendix at the end of the article is still rudimentary, and the only additional condition added is the absolute convergence of the series in question, which is quite natural. Unfortunately, in the eyes of mathematics journals such as the Annals of Mathematics (the leading mathematics journal on par with PRL), there is a lack of rigor in the writing in some places, and one example is the writing. This may be the writing style of theoretical physicists, you must know that Mr. Yang Zhenning once famously said: "Modern mathematics books can be divided into two kinds, one is that you can't read a page, and the other is that you can't read a line." ”
I'm looking at Hardy & Wright (e.). m.Wright, 1906-2005) found that on page 237 of the book, theorem 270 [to my surprise, there are 460 theorems in this book in just over 400 pages!] Another book I read, The Problem of Algebraic Eigenvalues, by James HWilkinson (1919-1986), also British, has a larger book (662 pages) but lists only four numbered theorems. ], which is essentially the same as the generalized formula (cnx) that Professor Chen used to solve the three physics inverse problems:
If and only if (hw)
After proving the classical Möbius inversion formula (i) and the generalized form of the actual variable situation ( ) in the book, the above two authors are impatient and simply give the reader the task of proving theorem 270: the reader should h**e no difficulty in constructing a proof with the help of theorem 263; but some care is required about convergence.(The reader should have no difficulty constructing a proof with the help of Theorem 263; But attention needs to be paid to convergence. Encouraged by this, I spread out my paper and did the exercises they assigned, and found that it was almost exactly the same as the proof ( ). Next, I will use the same method to prove the homework (HW) problem (HW) verification (cnx):
The second equation above is true because the infinite square matrix of numbers has the following method of grouping and rearranging the summation:
Among them, the order. In this way, one with infinite plurality is used instead of a finite sum. Naturally, as I said above, the absolute convergence of the relevant series needs to be assumed beforehand. Here, the condition is:
where d(k) is the number of positive factors of k.
Since the Möbius function is the Dirichlet inverse of the unit arithmetic function 1, the same reasoning proves a more "corrective" Möbius-type inversion formula than (cnx): for an arithmetic function with a Dirichlet inverse, if and only if (gcnx).
and its equivalent form.
If and only if (ghw)
Hopefully, these two formulas will also find applications to the physical sciences.
Academician Chen Nanxian is not only creative in research, but also enthusiastic in writing for the public. I was so unheard of that I didn't know his name until 2020, when I read a beautiful essay he wrote for Mathematical Culture, "The End of His People and His Affairs". The "Mo Bi" in the title is the translation of M Bius that he adopted, and the reason is in the first paragraph of the article: "The author thinks that Mr. Wang Zhuxi (1911-1983), who is well versed in the pronunciation of English and German, has the best translation: Mo Bi." "I marvel: physicists are different! At the same time, he also wondered: why does he have a soft spot for Mobi? Now it dawns on me: 30 years later, the German mathematicians have found a bosom friend of the Chinese physicist a century and a half later!
Because Professor Chen Nanxian's PRL** sounded the clarion call to plant the banner of number theory on the top of the physics mountain, the world's top journal "Nature" naturally paid special attention to him that year. The editor-in-chief, Sir John Royden Maddox (1925-2009), wrote a one-page commentary in the March 1990 edition of the 344th volume of News and Views, which presupposed: "Who said that number theory is purely academic and has nothing to do with practicality? The ancient Möbius theorem, which unexpectedly proved to be useful for solving physical inversion problems, may have important applications (who says that the theory of numbers is strictly academic?). an old theorem due to mobius has unexpectedly proved to be a way of solving physical problems of inversion that may h**e important applications)。The review praised Chen for "putting the Möbius inversion theorem into practice through his clever application" and cited three examples of practice that the creative physicist detailed in his PRL**. At the end of the review, the editor-in-chief felt that it was reasonable to guess that "Chen's testimony suggests that even Morbius is keeping the mysteries of the modern world at bay, and that there will now be a small group of people searching through the mysteries of the modern world in the hope of finding other useful tools in a field that might have previously been mistaken for barren land." ( it is fair to guess that, with chen's proof that even mobius has something to tell the modern world, a small army will now be scouring the literature of the theory of numbers in the hope of finding other useful tools in what may h**e been unjustly regarded as a backwater.)”
He's right. The seeds of pure mathematics, whether timeless formulas or steaming theories, are likely to bear fruit as long as they are widely sown across the vast expanse of the physical world. Physicists, get more exposure to math! Mathematicians, make friends with physicists!
This article is supported by the Science Popularization China Star Program Project, produced by the Science Popularization Department of the China Association for Science and Technology, supervised by the China Science and Technology Press, Beijing Zhongke Galaxy Culture Media***
*: Back to the basics. Edited by wnkwef