Unitary cyclotomic polynomials Pieter Moree and L´aszl´oT´oth November 6, 2019 Abstract The notion of block divisibility naturally leads one to introduce unitary cyclotomic poly- nomials. 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 M¨obius 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 M¨obius 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.