Degree Project Möbius Inversion Formula and Applications to Cyclotomic Polynomials 2012-06-01 Author: Zeynep Islek Subject: Mathematics Level: Bachelor Course code: 2MA11E Abstract This report investigates some properties of arithmetic functions. We will prove M¨obiusinversion formula which is very important in number theory. Also our report investigates roots of unity and cyclo- tomic polynomials over the complex numbers. 2 Contents 1 Introduction 4 2 Arithmetic Functions 7 2.1 The M¨obius Function . 7 2.2 The Euler Function . 10 2.3 M¨obiusInversion Formula . 13 3 Cyclotomic Polynomials and Roots of Unity 16 3.1 nth Root of Unity . 16 3.2 Cyclotomic Polynomials . 18 4 Conclusion 23 3 1 Introduction The number theory is a very important part of mathematics. We can say that it is a basis of mathematics. Carl Freidrich Gauss who is a very famous mathematician, was said that : `Mathematics is the queen of science and arithmetics is the queen of mathematics.' The classic M¨obiusfunction is an important multiplicative function in the number theory and combinatorics. In number theory there are two very important multiplicative functions which are M¨obiusfunction and Euler's function, denoted by µ and ' respectively. The M¨obius µ introduced to solve a problem which was related to Riemann zeta function. The Euler's function ' introduced to generalize a congruence result of Fermat but in this study we are not interested in congruence relating and Riemann zeta function. The classic M¨obiusfunction is defined by the August Ferdinant M¨obius, in 1832. It is a function whose domain is the positive integers, and which is defined as follows: 8 < 1 if n = 1 µ(n) = 0 if n is divisible by a square bigger than 1 : (−1)k if n is product of k distinct primes Dedekind and Liouville reported the inversion theorem for sums, simul- taneously. In 1857 they gave some appliction to '(m). R.Dedekind redeploid the function in the reverse of series which is given by M¨obius. E.Laguerre described the function below, using the function which is written by Dedekind. If X F (m) = f(d) where d ranges over the divisors of m, then X m f(m) = µ( )F (d): (1) d djm When these formulas and (1) were used, this formula was gotten: X '(d) = m: F. Mertens noted that if n > 1, P µ(d) = 0 where d ranges over the divisors of n. A.F. M¨obiusrecognized its arithmetical importance, in 1832. M¨obiusanaylsed the inverse of f which is an arbitrary function, using the Dirichlet series. Liouville and Dedekind gave the finite form of the M¨obius inversion formula, in 1857, as follows X X n g(n) = f(d) () f(n) = µ(d)g( ): d djn djn 4 If you want to learn more details about history of these functions, please read [1]. Now we will mention relationship between M¨obiusfunction and roots of unity, but before we will give some definitions. A polynomial, not identically zero, is said to be irreducible if it cannot be written as a product of two or more non-trivial polynomials whose coeffi- cients are of specified type. If you want to learn more, please check [2, 3, 4]. Every non-zero polynomial over C can be factored as p(x) = α(x − z1) ::: (x − zn) where n is the degree, α is the leading coefficient and z1; : : : ; zn the zeroes of p(x). If α 2 C and rational numbers c1; : : : ; cn exist satifying n n−1 α + c1α + ··· + cn = 0 then α is called an algebraic numbers. If you want to read more details, please check [3, 5, 6, 7]. Let's take any algebraic number α. The minimal polynomial of α is the unique irreducible polynomial of the smallest degree p(x) with rational co- efficients such that p(α) = 0 and whose leading coefficients 1. If you want to learn more details about minimal polynomials, please read [8]. In mathematics, a root of unity, is any complex number that equals 1 when raised to some integer power n. Roots of unity is important in number theory. An nth root of unity, is a complex number z satisfying the equation zn = 1 where n = 1; 2; 3;::: is a positive interger. 2πik An each element of this sum, which is shown e n implies that nth root 2πi k−n and k for the kth power. Equivalentely, we can use (e n ) instead of 2πik e n , in the complex plane. An nth root of unity is primitive if it is not a kth root of unity for some smaller k: zk 6= 1, k = 1; 2; : : : ; n − 1. The zeroes of the polynomial p(z) = zn − 1 are the nth roots of unity, each with multiplicity 1. There is a unique monic polynomial Φn(x) having degree '(n) whose root are the distinct primitive nth roots of unity, where ' is an Euler's function. Φ is called a cyclotomic polynomial. Finally, we can say that about this study. In the first part of this the- sis, we emphasize the M¨obiusfunction and we prove the M¨obiusinversion formula. Using this M¨obiusinversion formula, we prove some theorems. In 5 the second part we emphasize the roots of unity in the complex numbers, correspondingly we emphasize the cyclotomic polynomials in the complex numbers, and we will see the connection between the M¨obiusinversion for- mula and the cyclotomic polynomials in the complex numbers. 6 2 Arithmetic Functions In this section we describe M¨obius function and M¨obiusinversion formula, then we prove these functions. Also we prove some theorems which we need for proving M¨obius inversion formula. Now we start to give some definition about the number theory. Definition 2.1. A real or complex valued function defined on the positive integers is called an arithmetic function. In set notation: + f : Z −! R or + f : Z −! C If you want to see more details on arithmetic functions, check [9]. Definition 2.2. An arithmetic function f is called multiplicative if f(mn) = f(m)f(n) where m and n are relatively prime positive integers (i.e. (m; n) = 1). Definition 2.3. An arithmetic function f is called completely multiplicative if f(mn) = f(m)f(n) for every positive integers m and n. 1 + Example 1. The function f(x) = x arithmetic function where f : Z −! + R, because since all x in Z , the results are in R. Let's take x = 2, then 1 1 1 f(x) = x = 2 , and 2 is in R. Now it is necessary to learn next definition for continue apprehensibly. Definition 2.4. Let a; b 2 Z and a 6= 0 such that b = ax if there exist x 2 Z, then we say that \a divides b" which can be denoted ajb, and ajb if and only if b = ax for all x in Z. 2.1 The M¨obiusFunction The arithmetic function µ(n), defined for all natural numbers, is called M¨obiusfunction. Definition 2.5. The M¨obiusfunction µ(n) is defined as follows 8 < 1 if n = 1 µ(n) = 0 if n is divisible by a square larger than 1 k : (−1) if n = p1 : : : pk where pi's are relatively prime numbers 7 Example 2. We have µ(1) = 1 it is clear to see from the definition. More- over µ(2) = −1 because 2 is a prime number. So µ(2) = (−1)1 = −1. We have µ(4) = 0 because 22j4 or we can say that 4 is divisible by a square. We have µ(8) = 0 because 22j8. We have µ(42) = −1 because 42 = 2 · 3 · 7, so 42 can be written as the product of three relatively prime numbers. Thus µ(42) = (−1)3 = −1. Theorem 2.1. The function µ(n) is multiplicative. Proof. We will prove that µ(mn) = µ(m)µ(n) whenever m and n are rela- tively prime numbers. First, we consider m and n are square-free numbers. We assume that m = p1 : : : pk, where p1; : : : ; pk are distinct primes, and n = q1 : : : qs, where q1; : : : ; qs are distinct primes. From the definition of µ(n), k s we write that µ(m) = (−1) and µ(n) = (−1) , and mn = p1 : : : pkq1 : : : qs, again using the definition of µ(n), we write µ(mn) = (−1)k+s. Hence µ(mn) = (−1)k+s = (−1)k(−1)s = µ(m)µ(n): Now suppose at least one of m and n is divisible by a square of a prime, then mn is also divisible by the square of a prime. So µ(mn) = 0 and µ(m) or µ(n) is equal to zero. Now it is clear to see that the product of µ(m) and µ(n) is equal to zero. So µ(mn) = µ(m)µ(n). On the other hand, from the definition of µ(n), we know that µ(4) = 0 because 22j4 and µ(2) = −1. We can write that µ(4) = µ(2 · 2), but µ(4) = 0 6= µ(2)µ(2). Hence µ(n) is not completely multiplicative func- tion. The M¨obiusfunction appears in many different places in number theory. One of its the most important properties is a formula for the divisor sum P djn µ(d) , extended over the positive divisor of n. It leads to M¨obiusin- version formula. 8 Theorem 2.2. For the M¨obiusfunction µ(n), the summatory function is defined by X 1 if n = 1 µ(d) = 0 if n > 1 : djn We need the following theorem to prove Theorem 2.2 .