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The inverse Fourier transform is Company the convolution of f and g. There is also a similar convolution theorem for the commonly used Laplace transform in engineering mathematics. So, is the idea and Iron Foil 0.03mm Sheet 0.03mm Stainless Steel Foil 0.03mm Copper Foil 0.03mm Aluminum Foil 0.03mm, method of convolution also related to "Mobius inversion"? Of course there is! This is the Dirichlet convolution used for arithmetic functions in number theory, and this concept is simply a direct extension of Mobius inversion. Its definition is very similar to the expression at the right end of the Mobius inversion formula (I), except for the Mobius function there μ Substituted by general functions: if f and g are arithmetic functions, then the Dirichlet convolution of f and g is an arithmetic function. Clearly, the Dirichlet convolution is also commutative like integer multiplication. In addition, under the addition of functions and multiplication with the connotation of Dirichlet convolution, all arithmetic functions form a commutative ring like all integers, called the Dirichlet ring. The multiplication unit of an integer ring is a positive integer 1, while the multiplication unit of a Dirichlet ring is the arithmetic function mentioned earlier ε， Its official name Nickel Foil 0.03mm Brass Foil 0.03mm Titanium Foil 0.03mm Niobium Foil 0.03mm Zinc Foil 0.03mm Tantalum Foil is "identity arithmetic function" and it has an identity. Of course, it's not a coincidence. In fact, using the same approach as proving the Mobius inversion formula above, we can quickly verify f

In addition, Dirichlet convolution, Company like integer multiplication, satisfies the associative and distributive laws: (f * g) * h=f * (g * h) and f * (g+h)=f * g+f * h. As for the Dirichlet ring, if and only Tantalum Foil 0.04mm Iron Foil 0.04mm Sheet 0.04mm Stainless Steel Foil 0.04mm Copper Foil 0.04mm, if the arithmetic function f satisfies f (1) ≠ 0, it has a Dirichlet inverse, that is, there exists an arithmetic function f-1 such that f * f-1= ε。 Specifically, the Dirichlet inverse of constant function 1 is the Mobius function μ， The relationship required in the next paragraph of the argument is 1* μ=ε。 Here we have used 1 to represent a function that takes a value of 1 everywhere on the set of natural numbers, and its name is "unit arithmetic function". With the powerful tools of convolution, we can provide a more concise proof of the Mobius inversion formula. Firstly, the formula (*) can be written in convolutional form g=f * 1. Furthermore, the derivation process of the inversion formula (I) from right to left is reversed. Under the condition that (I) holds, the following steps are taken to Aluminum Foil 0.04mm Nickel Foil 0.04mm Brass Foil 0.04mm Titanium Foil 0.04mm Niobium Foil 0.04mm Zinc Foil deduce that (*) is true: it can be seen that in the context of Dirichlet convolution, the expression of the classical Mobius transform is: g=f * 1 if and only if f=g* μ。 Generally, college students majoring in science and engineering usually start from Fourier series or

In the boundary value problem of Company partial differential equations, I learned the name of German mathematician Gustav Lejeune Dirichlet (1805-1859), but don't mistake it for thinking that he only specializes Zinc Foil 0.05mm Tantalum Foil 0.05mm Iron Foil 0.05mm Sheet 0.05mm Stainless Steel Foil 0.05mm, in "analytical mathematics", just like almost all mathematicians today are only proficient in one skill. He is also a master of number theory, pioneering the branch of analytical number theory. The modern definition of a function also stems from it, allowing today's global middle school students to benefit from this most reasonable definition. Since the Mobius inversion is only the Dirichlet inverse of unit arithmetic function 1, it is the Mobius function μ” The synonyms for this fact, the original Mobius transformation double formulas (*) and (I), can be immediately generalized into the following general inversion formula: assuming an arithmetic function α If there is a Dirichlet inverse, then only if the simpler and more prominent convolution form of (#) (#) is g= α* If and only if f= α- 1 * g. If we imagine the convolutional symbol as an arithmetic multiplication sign known to elementary school students, it would be as simple as 6=2 * 3 if and only if 3=2-1 * 6. It Copper Foil 0.05mm Aluminum Foil 0.05mm Nickel Foil 0.05mm Brass Foil 0.05mm Titanium Foil 0.05mm Niobium Foil can be seen that abstract mathematics is not so difficult to understand. We provide another extension of the classical Mobius inversion formula, which generalizes the arithmetic function defined on the natural number set to

Complex valued functions. To show the Company difference from the previous domain, the functions here will be represented in uppercase letters. Let F and G be two functions that map [1, ∞) onto a complex set, Niobium Foil 0.06mm Zinc Foil 0.06mm Tantalum Foil 0.06mm Iron Foil 0.06mm Sheet 0.06mm, satisfying equation (* *) where [x] is the maximum natural number less than or equal to x. We will deduce the following inversion formula (II). In fact, we only need to use the same method as proof (I) to deduce from the right end of (II) to the left end: the second equal sign above is because grouping is done according to mn=k, and the sum order is rearranged. The generalized form of the general formulas (#), (* *), and (II) corresponding to the discrete case is: if and only if the Euler function, since the classical Mobius inversion formula is derived from number theory, it seems unreasonable not to provide a specific application of it in number theory. We will choose the well-known Euler function in number theory φ Give an example. This function was introduced by Leonhard Euler (1707-1783) in 1763, Stainless Steel Foil 0.06mm Copper Foil 0.06mm Aluminum Foil 0.06mm Nickel Foil 0.06mm Brass Foil 0.06mm Titanium Foil and its value at the natural number n φ (n) Defined as the number of natural numbers that are not greater than n and coprime with n. The first ten Euler function values are φ (1) =1, φ (2) Let's calculate φ What is (pm) equal to, where p is a prime number. The maximum common denominator between pm and the natural numbers from 1 to pm

The factors can only be 1, p, p2,..., pm, Company so the natural numbers with a maximum common factor greater than 1 are p, 2p, 3p,..., pm-1p=pm, and there are a total of pm-1. The remaining natural numbers are prime to pm, Titanium Foil 0.07mm Niobium Foil 0.07mm Zinc Foil 0.07mm Tantalum Foil 0.07mm Iron Foil 0.07mm, so there is a formula φ (pm)=pm pm-1. The Euler function is integrative, meaning that for any coprime natural number m and n, there is φ (mn)= φ (m) φ (n) . We only provide a brief proof for m=pr and n=qs, where p and q are distinct prime numbers, which can generally be proven using the same method. From the previous paragraph, it can be seen that in natural numbers smaller than pr, there are φ (pr)=pr pr-1 coprimes with pr, denote their set as P; Similarly, in natural numbers smaller than qs, there are φ (qs)=qs qs-1 coprimes with qs, denote their set as Q. According to the "Chinese remainder theorem", the product P × Q of these Sheet 0.07mm Stainless Steel Foil 0.07mm Copper Foil 0.07mm Aluminum Foil 0.07mm Nickel Foil 0.07mm Brass Foil two sets has a one-to-one correspondence with all natural numbers that are not greater than prqs and are prime to each other. In other words, given the number a in P and the number b in Q, the number ab

The prime factorization of natural numbers is Company based on the product of Euler functions, which is the famous Euler product formula. The Chinese remainder theorem used above, also known as the Sun Tzu theorem, has Brass Foil 0.08mm Titanium Foil 0.08mm Niobium Foil 0.08mm Zinc Foil 0.08mm Tantalum Foil 0.08mm, nothing to do with the Sun Tzu Art of War. In the "Sun Tzu Suan Jing" during the Northern and Southern Dynasties period, there is a calculation question: "If something is unknown, there is two left for the number of three or three, three left for the number of five or five, and two left for the number of seven or seven. What is the geometry of things?" The decimal answer to this question is 23, and the theorization of its solution becomes the "Sun Tzu Theorem" for a system of linear congruence equations with one variable. Here is a special version of only two equations: let the integers m and n be coprime. For any integer a and b, the congruence equation system has a solution x=adn+bcm, where the integers c and d satisfy cm+dn=1. Now let's sum the factors of the Euler function:. What is it equal to? When n=pr, then when n=prqs, each factor of n can be written as the product of the factor of pr and the factor of qs. Therefore, there is a general case of right, Iron Foil 0.08mm Sheet 0.08mm Stainless Steel Foil 0.08mm Copper Foil 0.08mm Aluminum Foil 0.08mm Nickel Foil proving that it is essentially the same. The proven formula was established by Carl Friedrich Gauss (1777-1855). If we use the Dirichlet convolution form, it would be φ* 1=Id, where Id is an identity function. The Mobius inversion formula is now available. Apply (I) to

The Gaussian equation above provides a clue Company for finding the Dirichlet inverse of the Euler function, which is expressed explicitly as a Mobius function. Remove φ (n) The n in the expression, and then the d in Nickel Foil 0.09mm Brass Foil 0.09mm Titanium Foil 0.09mm Niobium Foil 0.09mm Zinc Foil 0.09mm, the denominator is moved up to the numerator to define an arithmetic function. Then, therefore. We will use the Mobius inversion formula for a class of polynomials with circular polynomials. The complex solution of the polynomial equation zn-1=0 is called the n-th root of 1, and they are denoted. Therefore, the n-th roots can be written to the power of, where the roots of the power k and n are prime to each other are called the n-th primitive roots of 1. An equivalent definition of a primitive root is that it is the root of zn-1, but not the root of any lower degree polynomial zm-1. The primitive root has the property of implying n | l. Given n, the root is exactly the first polynomial of all primitive roots Φ N is called an n-order cyclotomic polynomial, where gcd (k, n) represents the greatest Tantalum Foil 0.09mm Iron Foil 0.09mm Sheet 0.09mm Stainless Steel Foil 0.09mm Copper Foil 0.09mm Aluminum Foil common factor of k and n. The minimum natural number d that holds the equation zd=1 for any n-th root z of 1 is called the order of z, which satisfies d | n. This property can be used to prove the polynomial equation (* * *) below. In fact, if z is a zero point of the right-hand polynomial in (* * *), then for a positive factor of n

There is zd=1. If n=dm, then zinc=(zd) m=1m=1. On Company the other hand, if z=1, then the order d of z is divided by n, so z is the primitive d-th root of 1, that is, it is a cyclotomic polynomial Φ A zero point Copper Foil 0.1mm Aluminum Foil 0.1mm Nickel Foil 0.1mm Brass Foil 0.1mm Titanium Foil 0.1mm, of d. The (* * *) formula has been proven. Applying the multiplication form of the Mobius inversion formula (MI) to (* * *) yields an explicit expression for the circular polynomial: the infinite series up to this point are all finite series, that is, the sum of finite numbers. Next, consider several infinite series. When performing the Mobius inversion surgery on them by grouping and rearranging the series terms, it is necessary to ensure that the operation is correct. A sufficient condition for the success of the surgery is the absolute convergence of the relevant series. Once the infinite series are released, this assumption will be given without explanation. The reason is simple: only conditionally convergent series can rearrange the sequence of general terms so that the new series changes its sum. Niobium Foil 0.1mm Zinc Foil 0.1mm Tantalum Foil 0.1mm Iron Foil 0.1mm Sheet 0.1mm Stainless Steel Foil Let's first consider a class of special series named after the surname of the polymath Johann Heinrich Lambert (1728-1777). For the infinite sequence {f (n)}, assuming | x |<1, using the sum formula of proportional series, the left end of the above equation is called the Lambert series, and the right end is

Explain that it is equal to a power series, Company where {f (n)} and {g (n)} satisfy (*). Specifically, if there is an identity, then replacing x=e-z with a variable in the Lambert series formula yields Sheet 0.11mm Iron Foil 0.11mm Tantalum Foil 0.11mm Zinc Foil 0.11mm Niobium Foil 0.11mm, another form of it: a similar approach can be used for the so-called Dirichlet series. Multiplying the series expression of the Riemann function by the Dirichlet series, using the same technique as the Lambert series, has a special feature of taking f (n) as the Mobius function μ (n) Because (see previous Dirichlet convolution equation 1)* μ=ε）， The series expression of the reciprocal of the function can be derived using the proof method of equation ($), which leads to more general equations. Unexpectedly, the Mobius inversion formula and its application have not crossed the purely mathematical domain. Can't it find applications in other disciplines? Number theory was once considered an extremely pure branch of mathematics with great aesthetic value. At least the British number theory guru Godfrey Titanium Foil 0.11mm Brass Foil 0.11mm Nickel Foil 0.11mm Aluminum Foil 0.11mm Copper Foil 0.11mm Stainless Steel Foil Harold Hardy (1877-1947) firmly believed that only "low-level mathematics" such as calculus could be played with by applied scientists. Number theory, on the other hand, was regarded by the mathematical prince Gauss as the "queen of mathematics" who could only appreciate its beauty but could not be assigned

The global physics community seems to have not Company really put much effort into the Mobius inversion formula. Until 1990, the top physics journal, Physical Review Letters (PRL), published a paper signed Stainless Steel Foil 0.12mm Sheet 0.12mm Iron Foil 0.12mm Tantalum Foil 0.12mm Zinc Foil 0.12mm, solely by a Chinese author in Volume 64, Issue 11, which even alarmed the editor in chief of Nature at the time and resulted in a full page review. This Chinese scholar's name is Chen Nanxian (1937-). He graduated from the Department of Physics at Peking University and obtained a Ph.D. in Electrical Engineering and Science from the University of Pennsylvania in 1984. He was elected an academician of the CAS Member seven years after the publication of this novel. The title of the article is "Modified M ö bius inverse formula and its application in physics" (What is the modified M ö bius inverse formula? Its biggest difference in form from classical formulas is that the new formula is a pair of infinite series expressions, and the latter is the inverse representation of the Niobium Foil 0.12mm Titanium Foil 0.12mm Brass Foil 0.12mm Nickel Foil 0.12mm Aluminum Foil 0.12mm Copper Foil former (to be consistent with the symbols in this article, I changed A to G, B to F, and so on) ω Change to x): If and only if. At first glance, the differences between the above formula and the previous formulas (* *) and (II)

Only infinite series replaced finite series. It is Company the leap from poverty to infinity that easily makes people feel that proof is not so "elementary", and extreme thinking must be inserted to 0.13mm Stainless Steel Foil 0.13mm Sheet 0.13mm Iron Foil 0.13mm Tantalum Foil 0.13mm, help. The fact is, the proof provided by the author in the appendix at the end of the article is still elementary, and the only additional condition added is the absolute convergence of the series, which is very natural. Unfortunately, in the eyes of numerous mathematical journals such as Annals of Mathematics (a top mathematical journal on par with PRL), there is a lack of sufficient rigor in the writing of certain parts of the proof. One example is that it was written. This may be the writing style of theoretical physicists, but it should be noted that Mr. Yang Zhenning once said, "Modern mathematics books can be divided into two types: one is unable to read Copper Foil Zinc Foil 0.13mm Niobium Foil 0.13mm Titanium Foil 0.13mm Brass Foil 0.13mm Nickel Foil 0.13mm Aluminum Foil a page, and the other is unable to read a line." I was flipping through Hardy and Wright's (E. M. Wright, 1906-2005) masterpiece "Introduction to Number Theory To my surprise, this book only has over 400 pages and contains 460 theorems, while another book I have read, Algebraic Eigenvalue Problems, was written by James H. Wilkinson

Also from England, the book has a larger Company section (page 662), but only lists four labeled theorems, The extended formula (CNX) used by Professor Chen to solve three physical inverse problems is essentially Aluminum Foil 0.14mm Copper Foil 0.14mm Stainless Steel Foil 0.14mm Sheet 0.14mm Iron Foil 0.14mm, the same: if and only if. After proving the classical Mobius inversion formula (I) and its extended form in the case of real variables (II) in the book, the two authors mentioned above became impatient. As for Theorem 270, they simply assigned the reader the task of proof: Tt conversion (The reader should have no difficulty in constructing the proof with the help of Theorem 263; however, attention should be paid to convergence.) Encouraged by this, I spread out the paper to do the exercises they arranged, and found that it was almost the same as the proof (II). Below, I will use the same method as proving homework (HW) questions to verify (CNX): the second equation above is true because the infinite matrix of numbers has the following grouping and summing method: where. This eliminates the need for Tantalum Foil 0.14mm Zinc Foil 0.14mm Niobium Foil 0.14mm Titanium Foil 0.14mm Brass Foil 0.14mm Nickel Foil an infinite number of ∑ and replaces it with a finite sum. Naturally, as I mentioned earlier, it is necessary to assume the absolute convergence of the relevant series beforehand. Here, the condition is that d (k) is the number of positive factors of k. Since Mobius

Function μ It is the Dirichlet inverse of the Company unit arithmetic function 1, and similarly, it can be proven that the Mobius type inversion formula, which is more "modified" than (CNX), is applicable to Nickel Foil 0.15mm Aluminum Foil 0.15mm Copper Foil 0.15mm Stainless Steel Foil 0.15mm Sheet 0.15mm, arithmetic functions with Dirichlet inverse α， If and only if (GCNX) and its equivalent form are. (GHW) hopes that these two formulas can also find applications in physical science. Academician Chen Nanxian not only has creativity in research, but also is enthusiastic about writing for the public. I was ignorant and ignorant, and it wasn't until 2020 that I read his beautiful article "The End of the World" for "Mathematical Culture" that I first learned about his name. The "Last Bi" in the title is the M ö bius translation he adopted, and the reason is in the first paragraph of the text: "I think Mr. Wang Zhuxi (1911-1983), who is well versed in English and German pronunciation, translated it the best: Last Bi." I exclaim: Physicists are different! At the same time, he also wondered why he had a particular fondness for Mo Bi Qing? Now I suddenly realize: thirty years ahead, German mathematicians and sages had already found a Chinese physicist's confidant a century and a half later! Iron Foil 0.15mm Tantalum Foil 0.15mm Zinc Foil 0.15mm Niobium Foil 0.15mm Titanium Foil 0.15mm Brass Foil Due to Professor Chen Nanxian's PRL paper sounding the horn of placing the number theory flag on the top of physics, the world's top journal Nature paid special attention to him that year. Editor in Chief Maddox (Si

In the 34th volume of the monthly publication Company of the News and Views column, a page of commentary was written. The preface of the article was: "Who says number theory is purely academic and not Brass Foil 0.16mm Nickel Foil 0.16mm Aluminum Foil 0.16mm Copper Foil 0.16mm Stainless Steel Foil 0.16mm, practical? The ancient Mobius theorem was unexpectedly proven to be useful for solving physical inversion problems and may have important applications." And listed three practical examples detailed by this creative physicist in his PRL paper. At the end of the comment, the editor in chief felt that it was reasonable to speculate that Chen's proof suggests that even Mobius preserves the mysteries of the modern world, and now there will be a small group of people searching in number theory literature, Sheet 0.16mm Iron Foil 0.16mm Tantalum Foil 0.16mm Zinc Foil 0.16mm Niobium Foil 0.16mm Titanium Foil hoping to find other useful tools, which may have been mistaken for a barren land before. Whether it is a timeless classic formula or a hot and fresh theory, as long as it is widely scattered on the vast land of the physical world, it is possible to bear fruitful results. Physicists, get more exposure to mathematics! Mathematicians, make friends with physicists! Many 80s and 90s generation friends may still remember this ball. Just flip over the early computer mouse