Unitary Cyclotomic Polynomials

Unitary cyclotomic polynomials Pieter Moree and LászlóTóth November 6, 2019 Abstract The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution. Contents 1 Introduction 2 1.1 Unitarydivisors................................. .... 2 1.2 UnitaryRamanujansums . .. .. .. .. .. .. .. .. 2 1.3 Unitary cyclotomic polynomials . ........ 3 1.4 Inclusion-exclusion and Kronecker polynomials . ............. 4 2 Elementary properties of unitary cyclotomic polynomials 5 3 Unitary cyclotomic polynomials as products of cyclotomic polynomials 6 4 Further properties of unitary cyclotomic polynomials 8 4.1 Calculation of Φ˚ 1 ................................. 8 np˘ q 4.1.1 Some products involving the cos and sin functions . ......... 10 ˚ 4.1.2 Calculation of Φn atotherrootsofunity. 11 arXiv:1911.01743v1 [math.NT] 5 Nov 2019 4.2 Unitary version of Schramm’s identity . ......... 11 5 The coefficients of unitary cyclotomic polynomials 13 6 More on inclusion-exclusion polynomials 14 6.1 Unitary cyclotomics as inclusion-exclusion polynomials............... 16 7 Applications of Theorem 6 16 8 Connection with unitary Ramanujan sums 18 1 1 Introduction 1.1 Unitary divisors A divisor d of n (d, n N) is called a unitary divisor (or block divisor) if d, n d 1, P p { q “ notation d n (note that this is in agreement with the standard notation pa n used for prime a || a1 as || powers p ). If the prime power factorization of n is n p1 ps , then the set of its unitary b1 bs “ ¨ ¨ ¨ divisors consists of the integers d p1 p , where bi 0 or bi ai for any 1 i s. “ ¨ ¨ ¨ s “ “ ď ď The study of arithmetic functions defined by unitary divisors goes back to Vaidyanathaswamy [29] and Cohen [4]. For example, the analogs of the sum-of-divisors function σ and Euler’s totient function ϕ are σ˚ n d, respectively ϕ˚ n # j : 1 j n, j, n 1 , where p q“ d||n p q“ t ď ď p q˚ “ u ř j, n ˚ max d : d j, d n . p q “ t | || u Several properties of the unitary functions σ˚ and ϕ˚ run parallel to those of σ and ϕ, respectively. For example, both functions σ˚ and ϕ˚ are multiplicative, and σ˚ pa pa 1, p q “ ` ϕ˚ pa pa 1 for prime powers pa (a 1). The unitary convolution of the functions f and g p q“ ´ ě is defined by f g n f d g n d n N . p ˆ qp q“ p q p { q p P q d n ÿ|| The set of arithmetic functions f such that f 1 0 forms a commutative group under the p q‰ unitary convolution and the set of multiplicative functions is a subgroup. The identity is the function ǫ, given by ǫ 1 1, ǫ n 0 (n 1), similar to the case of Dirichlet convolution. p q “ p q “ ą The inverse of the constant 1 function under the unitary convolution is µ˚ n 1 ωpnq, where p q“p´ q ω n denotes the number of distinct prime factors of n. That is, p q n µ˚ µ˚ d ǫ n n N . (1) d “ p q“ p q p P q d n d n ÿ|| ´ ¯ ÿ|| See, e.g., the books by Apostol [1], McCarthy [14] and Sivaramakrishnan [21]. 1.2 Unitary Ramanujan sums The unitary Ramanujan sums c˚ k were defined by Cohen [4] as follows: np q c˚ k ζjk k,n N , np q“ n p P q 1ďjďn pj,nÿq˚“1 2πi{n where ζn : e . (The classical Ramanujan sums are defined similarly, but with j, n 1 “ p q˚ “ replaced by j, n 1.) p q“ The identities c˚ k dµ˚ n d n, k N , (2) np q“ p { q p P q d k,n ||pÿ q˚ ˚ n if n k; c k ̺n k | (3) d p q“ p q“ d n #0 otherwise, ÿ|| can be compared to the corresponding ones concerning the classical Ramanujan sums cn k . p q Note that c˚ n ϕ˚ n , c˚ 1 µ˚ n (n N). np q“ p q np q“ p q P 2 1.3 Unitary cyclotomic polynomials The cyclotomic polynomials Φn x are defined by p q n j Φn x x ζn . (4) p q“ j“1 ´ pj,nźq“1 ` ˘ They arise as irreducible factors (see Weintraub [30]) on factorizing xn 1 over the rationals: ´ n x 1 Φd x . (5) ´ “ p q d n ź| By Möbius inversion it follows from (5) that µpdq µpn{dq n{d d Φn x x 1 x 1 , (6) p q“ ´ “ ´ d n d n ź| ´ ¯ ź| ´ ¯ where µ denotes the Möbius function. The unitary cyclotomic polynomial Φ˚ x is defined by np q n ˚ j Φn x x ζn , (7) p q“ j“1 ´ pj,nźq˚“1 ` ˘ see [21, Ch. X]. It is monic, has integer coefficients and is of degree ϕ˚ n . Furthermore, for any p q natural number n we have xn 1 Φ˚ x (8) ´ “ dp q d n ź|| and µ˚pdq µ˚pn{dq Φ˚ x xn{d 1 xd 1 . (9) np q“ ´ “ ´ d n d n ź|| ´ ¯ ź|| ´ ¯ See Section 2 for short direct proofs of these properties and further basic properties of unitary cyclotomic polynomials. If n is squarefree, then the unitary divisors of n coincide with the divisors of n and hence ˚ ˚ comparing (6) with (9) yields Φ x Φn x . In this case Φ x is irreducible over the np q “ p q np q rationals. However, a quick check shows, that for certain non-squarefree values of n, the ˚ ˚ polynomial Φn x is reducible over the rationals. For example, Φ12 x Φ6 x Φ12 x and ˚ p q ˚ p q “ p q p q Φ40 x Φ10 x Φ20 x Φ40 x . Indeed, we will show that Φ is reducible for every non-squarefree p q“ p q p q p q n integer n. This is a corollary of the fact that each polynomial Φ˚ x can be written as the product np q of the cyclotomic polynomials Φd x , where d runs over the divisors of n such that κ d κ n , p q p q“ p q with κ n the squarefree kernel of n (Theorem 2). In fact, this is a consequence of a more general p q result (Theorem 6) involving unitary divisors. One can introduce the bi-unitary cyclotomic polynomials Φ˚˚ x defined by n p q n ˚˚ j Φn x x ζn , p q“ j“1 ´ pj,nźq˚˚“1 ` ˘ 3 where j, n stands for the greatest common unitary divisor of j and n. The degree of the p q˚˚ polynomial Φ˚˚ x equals ϕ˚˚ n , the bi-unitary Euler function, which is defined as ϕ˚˚ n n p q p q p q “ # j : 1 j n, j, n 1 , see the paper [26]. Although these definitions seem to be more t ď ď p q˚˚ “ u natural than the previous ones, the properties of Φ˚˚ x and ϕ˚˚ n are not similar to their n p q p q unitary analogs. For example, the function ϕ˚˚ n is not multiplicative and the coefficients of ˚˚ p q ˚˚ 3 2 the polynomials Φ x are in general not integers (we have, e.g., Φ6 x x ηx ηx η, n p q p q“ ´ ` ` where η 1 i?3 2 and η 1 i?3 2). “ p ` q{ “ p ´ q{ 1.4 Inclusion-exclusion and Kronecker polynomials Let ρ r1, r2, . , rs be a set of increasing natural numbers satisfying ri 1 and ri, rj 1 “ t u ą p q“ for i j, and put ‰ n0 n0 n0 r , n , n i j ,... i i r ij r r “ i “ i “ i j r ‰ s ź For each such ρ we define a function Qρ by n0 nij x 1 iăj x 1 Qρ x p ´ q ¨ p ´ q¨¨¨ . (10) p q“ xni 1 xnijk 1 ip ´ q ¨ śiăjăkp ´ q¨¨¨ ś ś It can be shown that Qρ x is a polynomial of degree n0 1 1 ri having integer coeffi- p q ri|n0 p ´ { q cients. This class of polynomials was introduced by Bachman [2], who named them inclusion- ś exclusion polynomials. A Kronecker polynomial f Z x is a monic polynomial having all its roots inside or on P r s the unit circle. It was proved by Kronecker, cf. [6], that such a polynomial is a product of a monomial and cyclotomics and so we can write s ed f x x Φd x , (11) p q“ p q d ź with s, ed 0 and ed 1 for only finitely many d. ě ě We will show how a unitary cyclotomic can be realized as an inclusion-exclusion cyclotomic. As Qρ x is monic and in Z x , it follows from (10) that it is Kronecker. Thus we have the p q r s following inclusions: unitary cyclotomics inclusion-exclusion polynomials Kronecker polynomials . (12) t u Ă t u Ă t u The inclusion-exclusion polynomials that are unitary can be precisely identified (for the proof see Section 6.1). Theorem 1. The set of unitary polynomials Φ˚ x with n 2 equals the set of inclusion- np q ě exclusion polynomials Qρ x with ρ having prime power entries, with no base prime repeated.

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