Möbius Inversion Formula and Applications to Cyclotomic Polynomials

Degree Project

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 cyclotomic 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 Möbiusfunction is an important multiplicative function in the number theory and combinatorics. In number theory there are two very important multiplicative functions which are Möbiusfunction and Euler’s function, denoted by µ and ϕ respectively. The Möbius µ introduced to solve a problem which was related to Riemann zeta function. The Euler’s function ϕ 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 Möbiusfunction is defined by the August Ferdinant Möbius, in 1832. It is a function whose domain is the positive integers, and which is defined as follows:   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 ϕ(m). R.Dedekind redeploid the function in the reverse of series which is given by Möbius. 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 d|m When these formulas and (1) were used, this formula was gotten: X ϕ(d) = m.

F. Mertens noted that if n > 1, P µ(d) = 0 where d ranges over the divisors of n. A.F. Möbiusrecognized its arithmetical importance, in 1832. Möbiusanaylsed the inverse of f which is an arbitrary function, using the Dirichlet series. Liouville and Dedekind gave the finite form of the Möbius inversion formula, in 1857, as follows X X n g(n) = f(d) ⇐⇒ f(n) = µ(d)g( ). d d|n d|n

4 If you want to learn more details about history of these functions, please read [1]. Now we will mention relationship between Möbiusfunction 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 coefficients 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 ∈ 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 ϕ(n) whose root are the distinct primitive nth roots of unity, where ϕ 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 Möbiusfunction and we prove the Möbiusinversion formula. Using this Möbiusinversion 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 formula and the cyclotomic polynomials in the complex numbers.

6 2 Arithmetic Functions

In this section we describe Möbius function and Möbiusinversion formula, then we prove these functions. Also we prove some theorems which we need for proving Möbius 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 ∈ Z and a 6= 0 such that b = ax if there exist x ∈ Z, then we say that “a divides b” which can be denoted a|b, and a|b if and only if b = ax for all x in Z.

2.1 The MöbiusFunction The arithmetic function µ(n), defined for all natural numbers, is called Möbiusfunction.

Definition 2.5. The Möbiusfunction µ(n) is defined as follows   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 22|4 or we can say that 4 is divisible by a square. We have

µ(8) = 0 because 22|8. 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 relatively 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 22|4 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 function.

The Möbiusfunction appears in many different places in number theory. One of its the most important properties is a formula for the divisor sum P d|n µ(d) , extended over the positive divisor of n. It leads to Möbiusin- 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 . d|n We need the following theorem to prove Theorem 2.2 .

Theorem 2.3. If f is multiplicative function of n, and F is deﬁned as follows X F (n) = f(d) d|n then F is also multiplicative function.

Proof. We will show that F is multiplicative function. If F is multiplicative function, we write that when m and n are relatively numbers, then F (mn) = F (m)F (n). So, now let us choose (m, n) = 1. We have X F (mn) = f(d). d|mn

Now, all divisiors of mn must be written as the product of relatively prime numbers. As we mentioned before, if F is multiplicative function, we write F (mn) = F (m)F (n) when (m, n) = 1. So we write d = d1d2 as the product of relatively prime divisors d1 of m and d2 of n. Hence, we write X F (mn) = f(d1d2).

d1|m d2|n

Since f is multiplicative and since (d1, d2) = 1, we can write that X X X F (mn) = f(d1)f(d2) = f(d1) f(d2)

d1|m d1|m d2|n d2|n = F (m)F (n).

Now we continue to prove the Theorem 2.2.

Proof. Consider, n = 1. It is clear to see that X X µ(d) = µ(d) = µ(1) = 1. d|n d|1

9 Now we assume this formula for n > 1. Let us define an arithmetic function M as X M(n) = µ(d). d|n The Möbiusfunction is multiplicative, then M(n) is multiplicative by The- orem 2.3. Let’s suppose that n which is the product of powers of r different relatively prime numbers such that

r Y ai n = pi . i=1 Then the results which are under the function of M are equal. So we write

r Y ai M(n) = M(pi ). i=1 Now we are searching what is the result of M(n). If we find the result ai ai of M(pi ), we find the result of M(n). Now we can write that M(pi ) = P ai µ(d) using the Theorem 2.3, which is the way to find the result. We d|pi have

ai X M(pi ) = µ(d) ai d|pi 2 ai = µ(1) + µ(pi) + µ(pi ) + ··· + µ(pi ) = 1 + (−1) + 0 + ··· + 0 = 0.

For every integer bigger than 1, we proved that the sum function of M¨obius function is equal to zero.

The another example of an arithmetic function is Euler’s function. It is also a multiplicative function.

2.2 The Euler Function The function was introduced by Euler, in 1760, and is denoted by ϕ. This function is multiplicative which is one of the most important function in number theory.

Deﬁnition 2.6. The Euler function ϕ(n) is the number of positive integers less than n which are relatively prime to n.

10 Example 3. Here is some values of ϕ(n).

ϕ(1) = 1, ϕ(2) = 1, ϕ(3) = 2, ϕ(4) = 2, ϕ(5) = 4

ϕ(6) = 2, ϕ(7) = 6, ϕ(8) = 4, ϕ(9) = 6, ϕ(10) = 4

Now we ﬁnd the values of the phi-function at primes powers.

Theorem 2.4. Let p be a prime number. Then ϕ(pα) = pα − pα−1.

Proof. Between 1 and pα there are pα integers. There are some numbers which are not relatively prime to pα, they are p, 2p, . . . , pα−1. There are exactly pα−1 such integers. So there are pα − pα−1 integers less than pα that are relatively prime to pα. Hence, ϕ(pα) = pα − pα−1.

Theorem 2.5. If p is a prime, then ϕ(p) = p − 1.

Proof. It is easy to see that from Theorem 2.4. Now we suppose α = 1 then

ϕ(pα) = pα − pα−1 = p1 − p1−1 = p − 1.

Example 4. We calculate the values of ϕ(n) for some prime numbers.

ϕ(53) = 53 − 52 = 100

ϕ(210) = 210 − 29 = 512 ϕ(112) = 112 − 111 = 110

The Euler’s ϕ function is multiplicative function. If you’re interested in proof of ϕ is multiplicative function, please read [3, 5, 7].

Example 5. Let’s calculate ϕ(756) using that the Euler’s ϕ function is multiplicative. This number can ben be written as 756 = 22 · 33 · 7. Hence

ϕ(756) = ϕ(22 · 33 · 7).

We know that ϕ function is multiplicative. So we write

ϕ(756) = ϕ(22) · ϕ(33) · ϕ(7).

Using the Theorem 2.4 and Theorem 2.5, we ﬁnd

ϕ(22) = 22 − 2 = 2,

11 ϕ(33) = 33 − 32 = 18, ϕ(7) = 7 − 1 = 6. So ϕ(756) = 2 · 18 · 6 = 216. Now we get the result. If you want to check more examples, read [5]. P Theorem 2.6. For every positive integers d and n, we have d|n ϕ(d) = n. Proof. We will prove this by induction on the number of diﬀerent prime factors. We consider the case n = pα, where p is a prime number. We have X X ϕ(d) = ϕ(d) = ϕ(1) + ϕ(p) + ϕ(p2) + ··· + ϕ(pα) d|n d|pα = 1 + (p − 1) + (p2 − p) + ··· + (pα − pα−1) = pα = n.

Correspondingly, now suppose that the theorem holds for integers with k distinct prime factors. Let us take any integer N with k +1 distinct prime factors and pα be the highest power of p that divides N. Now we write N = pαn where p and n are relatively prime numbers (i.e., (p,n)=1). We know when d ranges over the divisor of n, the set d, dp, dp2, . . . , dpα ranges over the divisors of N. Then

X X X X X ϕ(d) = ϕ(d) + ϕ(dp) + ϕ(dp2) + ··· + ϕ(dpα) d|N d|n d|n d|n d|n X X X X = ϕ(d) + ϕ(d)ϕ(p) + ϕ(d)ϕ(p2) + ··· + ϕ(d)ϕ(pα) d|n d|n d|n d|n X = ϕ(d)[1 + ϕ(p) + ϕ(p2) + ··· + ϕ(pα)] d|n X X = ϕ(d) ϕ(e) d|n e|pα = npα = N. P We showed that d|n ϕ(d) = n is true for every positive integers n, and d ranges over n.

12 Example 6. We give an example to understand the Theorem 2.6. X ϕ(d) = ϕ(1) + ϕ(2) + ϕ(3) + ϕ(4) + ϕ(6) + ϕ(12) d|12 =1+1+2+2+2+4 = 12.

REMARK: As ϕ is an arithmetic function, we note that X X n ϕ(d) = ϕ( ). d d|n d|n Now we show that for any multiplicative functions: If f is multiplicative P P function and not equal to zero, then d|n f(d) or, equivalently, d|n f(n|d) denotes the sum of the values of a function f where d ranges over the positive divisors of n. We write that X X f(d) = f(n|d) d|n d|n

n because since d ranges over n, d ranges over n. For example; X f(d) = f(1) + f(2) + f(3) + f(6) + f(9) + f(18) d|18

X n 18 18 18 18 18 18 f( ) = f( ) + f( ) + f( ) + f( ) + f( ) + f( ) d 1 2 3 6 9 18 d|18 = f(18) + f(9) + f(6) + f(3) + f(2) + f(1)

2.3 M¨obiusInversion Formula Theorem 2.7. If g is any arithmetic function and f is the sum function of g, so that X f(n) = g(d) d|n then X n g(n) = f(d)µ( ). d d|n n Equivalently, if d ranges over n, d ranges over n. Hence, we can write X n X n f(d)µ( ) = f( )µ(d). d d d|n d|n

13 Proof. The equality X n X n f(d)µ( ) = f( )µ(d) d d d|n d|n is true from the remark. If d|n, we write n = ed from the deﬁnition (divisibility), where e is in Z. n Now let us take n = de, so e = d . Then the previous sum can be written as X f(d)µ(e) de=n and it is possible to write the last sum as, X f(e)µ(d). de=n P n Now we must prove that the sum d|n f(d)µ( d ) is equal to g(n) or, P n equivalentely, the sum d|n f( d )µ(d) is equal to g(n). Using equality below n X f( ) = g(e) d n e| d we write that X n X X µ(d)f( ) = (µ(d) g(e)). d n d|n d|n e| d n Since e divides d , then e divides n. Inversely, each divisor of n is e which divides n if and only if d divides n . So d divides n. As have seen, the d P e coeﬃcent of g(e) is n µ(n) can be written as d| e

X 1 if n = 1 µ(n) = e 0 if n > 1 n e d| e using the Theorem 2.2. That implies g(n) has only one coeﬃcient g(e) which is not equal to zero. So g(e) = 1. Then X n g(n) = f( )µ(d). d d|n

The Euler function is related to the M¨obiusfunction through the following formula.

14 Theorem 2.8. X µ(d) ϕ(n) = n d d|n where ϕ(n) is Euler’s ϕ function. P Proof. We know d|n ϕ(d) = n from the Theorem 2.6. Take a function F which is the sum of the Euler’s ϕ function is as X F (n) = ϕ(d) = n. d|n

Use the M¨obiusinversion formula here

X n ϕ(n) = F (d)µ( ) d d|n X n = F ( )µ(d) d d|n X n = µ(d) d d|n X µ(d) = n . d d|n

Example 7. Let’s calculate the value of ϕ(756), using Theorem 2.8. The divisors of this number are 1, 2, 3, 4, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, and 756. Now using the Theorem 2.8 we can calculate ϕ(756).

X µ(d) ϕ(756) = n d d|n X µ(d) = 756 · d d|756 X n = µ(d) d d|n µ(1) µ(2) µ(378) µ(756) = 756 · + + ··· + + 1 2 378 756

And here is some values of µ(n).

µ(1) = 1, µ(2) = −1, µ(3) = −1, µ(6) = 1,

15 µ(7) = −1, µ(14) = 1, µ(21) = 1, µ(42) = −1.

The value µ of the other numbers are equal to zero, because these numbers are divisible by a square larger than 1. Hence

1 −1 −1 1 −1 1 1 −1 2 756 · + + + + + + + = 756 · = 216 1 2 3 6 7 14 21 42 7

We solved same example using by ϕ function is a multiplicative function. That way is easy and short, because if you want to use the Theorem 2.8, you should know that the values of µ function.

3 Cyclotomic Polynomials and Roots of Unity

3.1 nth Root of Unity Assume that a be an nth root of a number b, this means that bn = a. In particular the square root of 1 is 1 because 1 · 1 = 1, but (−1)(−1) = 1, so −1 also a square root of 1. There are two square roots of 1. If we take the cube root of 1, then −1 is not a solution because (−1)(−1)(−1) = −1. So 1 has only one solution if we study on real numbers, but in the complex numbers, there are three roots of 1. All cube roots of 1 can also be deﬁned as powers of the negative interval. We see them below

16 REMARK: Let n be an integer and x be a complex number (and, in particular, a real number), the Euler’s function states that

ei(nx) = cos(nx) + isin(nx).

C be the field of complex numbers, there are exactly n different nth roots of 1. If you divide the unit circle into n equal parts, using n points, it is easy to find them.

Deﬁnition 3.1. A complex number z is called an nth root of unity for a positive integer n, if zn = 1.

We will show that zn = 1. Let’s take any complex number z. If we write this complex number on the polar coordinates, we get

z = cosθ + isinθ.

If we take the nth power of z,

zn = (cosθ + isinθ)n = cos(n · θ) + isin(n · θ).

We can write this equality, because this is de Moivre’s formula. So

zn = 1

cos(n · θ) + isin(n · θ) = cos(0) + isin(0).

17 2kπ Then nθ = 0 + 2kπ, where k = 0, 1, . . . , n − 1. Thus θ = n . The roots of unity is then in e2πi/n for k = 0, 1, . . . , n − 1. There are n diﬀerent solutions for zn = 1, namely, e2πi/n, e2πi2/n,..., 2πin/n 2πi/n 2 n e . We usually assume that ζn = e so that ζn, ζn, . . . , ζn are the nth roots of zn = 1.

k Deﬁnition 3.2. An nth root of unity is primitive if it is of the form ζn with k and n relatively prime numbers, i.e., (k, n) = 1. If ζn is a primitive nth k m root of unity and (ζn) = 1 then n|m. If you want to read more details, check [6].

3.2 Cyclotomic Polynomials k Deﬁnition 3.3. Let n be a positive integer and let ζn be the primitive 2πi/n nth root of unity ( ζn is the complex number e ). The nth cyclotomic polynomial Φn(x) is Y k Φn(x) = (x − ζn) 1≤k≤n gcd(n,k)=1 whose roots are the primitive nth roots of unity.

Theorem 3.1. Let n be a positive integer. Then

n Y x − 1 = Φd(x) d|n where d ranges over the divisor of n.

Proof. The roots of xn −1 are exactly nth roots of unity. On the other hand, if ζ is an nth root of unity and the order of ζ is d, then ζ is a primitive dth root of unity. So ζ is a root of Φd(x). But d|n, so ζ is a root of the right hand side. It follows that the polynomials on the left and right hand side have the same roots. Thus they are equal.

Another way to ﬁnd the nth cyclotomic polynomial: If n > 1, then xn − 1 Φn(x) = Q (2) d Φd(x) where d ranges over, except n, the divisor of n.

If you want to see more detail about cyclotomic polynomials, check [4, 10].

18 Example 8. Here is some value of cyclotomic polynomials.

Φ1(x) = x − 1

x2 − 1 x2 − 1 Φ2(x) = = = x + 1 Φ1(x) x − 1 3 3 x − 1 x − 1 2 Φ3(x) = = = x + x + 1 Φ1(x) x − 1 4 x − 1 2 Φ4(x) = = x + 1 Φ1(x)Φ2(x) 5 x − 1 4 3 2 Φ5(x) = = x + x + x + x + 1 Φ1(x) 6 x − 1 2 Φ6(x) = = x − x + 1 Φ1(x)Φ2(x)Φ3(x) n Q Now we know that x − 1 = d|n Φd(x), and conversely, by using the M¨obiusfunction, we can write the following theorem.

Theorem 3.2. Let n be a positive integer and µ(n) denotes the M¨obius function. Then Y d µ( n ) Φn(x) = (x − 1) d . d|n

n Q Proof. To prove this formula, first we use the equality x − 1 = d|n Φd(x), then we take complex logarithm of this equality, and finally use the Möbius inversion formula. We have

n Y x − 1 = Φd(x). d|n

Now take the complex logarithm both side of equality. It doesn’t eﬀect to the equality.   n Y log(x − 1) = log  Φd(x) d|n

Let’s assume d = {d1, d2, . . . , ds}, where di’s are divisor of n and are not equal to n.   Y log  Φd(x) = log (Φd1 (x) · Φd2 (x) ··· Φds (x)) d|n

19 Using the property of logarithm, we write

log (Φd1 (x) · Φd2 (x) ··· Φds (x)) = log(Φd1 (x)) + log(Φd2 (x)) + ··· + log(Φds (x)) X = log(Φd(x)). d|n

Using M¨obiusinversion formula, we can write X n log(Φ (x)) = µ( ) · log(xd − 1) n d d|n X d µ( n ) = log(x − 1) d d|n n n d µ( ) d µ( ) d µ( n ) = log((x 1 − 1) d1 ) + log((x 2 − 1) d2 ) + ··· + log((x s − 1) ds ) n n d µ( ) d µ( ) d µ( n ) = log (x 1 − 1) d1 · (x 2 − 1) d2 ··· (x s − 1) ds   Y d µ( n ) = log  (x − 1) d  d|n

If we cancel out the logarithm, we get the result

Y d µ( n ) Φn(x) = (x − 1) d . d|n

Now we can see the connection between M¨obiusinversion formula and cyclotomic polynomials. When

n Y x − 1 = Φd(x), d|n it is possible to write that

Y d µ( n ) Φn(x) = (x − 1) d . d|n

Example 9. We are going to ﬁnd cyclotomic polynomials using the formula

Y d µ( n ) Φn(x) = (x − 1) d d|n for n = 1, 2,..., 20. Φ1(x) = x − 1 Φ2(x) = x + 1 2 Φ3(x) = x + x + 1

20 2 Φ4(x) = x + 1 4 3 2 Φ5(x) = x + x + x + x + 1 2 Φ6(x) = x − x + 1 6 5 4 3 2 Φ7(x) = x + x + x + x + x + x + 1 4 Φ8(x) = x + 1 6 3 Φ9(x) = x + x + 1 4 3 2 Φ10(x) = x − x + x − x + 1 10 9 8 7 6 5 4 3 2 Φ11(x) = x + x + x + x + x + x + x + x + x + x + 1 4 2 Φ12(x) = x − x + 1 12 11 10 9 8 7 6 5 4 3 2 Φ13(x) = x + x + x + x + x + x + x + x + x + x + x + x + 1 6 5 4 3 2 Φ14(x) = x − x + x − x + x − x + 1 8 7 5 4 3 Φ15(x) = x − x + x − x + x − x + 1 8 Φ16(x) = x − 1 16 15 14 13 12 11 10 9 8 7 6 5 Φ17(x) = x + x + x + x + x + x + x + x + x + x + x + x + x4 + x3 + x2 + x + 1 6 3 Φ18(x) = x − x + 1 18 17 16 15 14 13 12 11 10 9 8 7 Φ19(x) = x + x + x + x + x + x + x + x + x + x + x + x + x6 + x5 + x4 + x3 + x2 + x + 1 8 6 4 2 Φ20(x) = x − x + x − x + 1

We have already found some cyclotomic polynomials in example 8. In this example you don’t need to know values of Möbiusfunction. If you want to find cyclotomic polynomials using the Theorem 3.2, you should know the values of Möbiusfunction. As have seen, the coefficients of cyclotomic polynomial are often −1, 0 and 1, but for n ≥ 105 some coefficients are different from this set. For example n = 105, then

48 47 46 43 42 41 40 39 36 35 34 Φ105(x) =x + x + x − x − x − 2x − x − x + x + x + x + x33 + x32 + x31 − x28 − x26 − x24 − x22 − x20 + x17 + x16 + x15 + x14 + x13 + x12 − x9 − x8 − 2x7 − x6 − x5 + x2 + x + 1.

We see that there is a coeﬃcient −2 which is not included in {−1, 0, 1}.

Theorem 3.3. The coeﬃcients of Φn(x) are integers.

Proof. We prove it using inductive method. Clearly, Φ1(x) = x − 1 ∈ Z[x]. Now we suppose that, for k < n, the coeﬃcients of Φn(x) are integers. Q Let f(x) = d|n Φd(x). Then we say that f(x) ∈ Z[x], from the inductive d

xn − 1 Φ (x) = n f(x)

21 then n x − 1 = Φn(x)f(x) n On the other hand, x − 1 ∈ Z[x]. Also using the division algorithm, n we can write x − 1 = f(x)g(x) + r(x) for some g(x), r(x) ∈ Z[x]. By the uniqueness, we take r(x) = 0. So xn − 1 = f(x)g(x). It is easy to see g(x) = Φn(x). Since g(x) ∈ Z[x], Φn(x) ∈ Z[x]. Finally we say that the coeﬃcients of Φn(x) are integers.

22 4 Conclusion

In this study, the M¨obiusinversion formula has been introduced and proved. The roots of unity and cyclotomic polynomials have been introduced.

In the first cheapter, some informations about Möbius function, Euler’s ϕ function, Möbiusinversion formula, roots of unity and cyclotomic polynomials have been introduced. And also some definitions which will be used in the next cheapters have been mentioned.

In the second cheapter, some arithmetic functions and properties of these functions have been studied. The Möbiusfunction which is defined on arithmetic functions, has been studied. The Euler’s ϕ function and its applications have been mentioned. The Möbiusinversion formula has been introduced. A theorem which is related with Euler’s ϕ function and µ function, using Möbiusinversion formula, has been proved, that is

X µ(d) ϕ(n) = n . d d|n

In the third cheapter, the roots of unity in the complex numbers have been introduced. Accordingly, the primitive roots of unity have been introduced. Later, the cyclotomic polynomials whose roots are the primitive nth roots of unity, have been investigated. Accordingly, the connection between M¨obiusinversion formula and cyclotomic polynomials has been shown. The connection between M¨obiusinversion formula and cyclotomic polynomials in the complex numbers has been shown. Since this is the formula below,

n Y x − 1 = Φd(x). d|n and with the M¨obiusinversion formula and using the complex logarithm

Y d µ( n ) Φn(x) = (x − 1) d d|n has been proved.

23 References

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