Unitary Cyclotomic Polynomials

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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 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. p q e1 es More precisely there is a one-to-one map between these sets that sends n to ρ p1 ,...,ps , e1 es e1 es “ t u where p1 p with p1 . . . p is the prime factorization of n, resulting in ¨ ¨ ¨ s ă ă s ˚ Φ x Q e1 es x . np q“ tp1 ,...,ps up q

4 This theorem shows that the first inclusion in (12) is strict, e.g., Qt5,6u x is not a unitary p q 2 cyclotomic. By Theorem 2 (or Theorem 26) any Kronecker polynomial divisible by Φd x for p q some d 1 cannot be an inclusion-exclusion polynomial, and so also the second inclusion is ě strict. Even more, it is easy to see that for both inclusions the set theoretic differences are infinite. We would like to point out that in this paper, with the exception of Theorem 22, the nomination “theorem” is not used to indicate a deep result, but rather a key fact.

2 Elementary properties of unitary cyclotomic polynomials

The polynomials Φ˚ x have integer coeﬃcients. This follows by induction on n by taking np q into account identity (8), similar to the case of classical cyclotomic polynomials. Indeed, various of our arguments in this section closely mirror those for cyclotomic polynomials and can, in somewhat more detail than we provided, be found in Thangadurai [24]. By the deﬁnition (7) and (1),

n n ˚ j j ˚ log Φn x log x ζn log x ζn µ d . p q“ 1 ´ “ 1 ´ p q j“ j“ d||pj,nq˚ pj,nÿq˚“1 ` ˘ ÿ ` ˘ ÿ

Note that d j, n holds if and only if d j and d n. Hence || p q˚ | || n{d log Φ˚ x µ˚ d log x ζk µ˚ d log xn{d 1 , np q“ p q ´ n{d “ p q ´ d n k“1 d n ÿ|| ÿ ´ ¯ ÿ|| ´ ¯ giving (9), which by unitary Möbius inversion is equivalent to (8). The unitary divisors of prime powers pa (a 1) are 1 and pa. We deduce by (8) that ě pa a ˚ x 1 Φ a x ´ Φ j x . (13) p x 1 p p q“ “ j“1 p q ´ ź From formula (9) we immediately see that the Taylor series of Φ˚ x around x 0 has np q “ integer coefficients, showing again that the coefficients of Φ˚ x have to be integers. np q Using (1), we see that, for n 1, we can rewrite (9) as ą µ˚pn{dq Φ˚ x 1 xd . (14) np q“ ´ d n ź|| ´ ¯ From (14) and (9) it follows that for n 1 ą ˚ Φ˚ x xϕ pnqΦ˚ 1 x , (15) np q“ np { q in other words, unitary cyclotomics are self-reciprocal. For odd n 1, we have ą ˚ ˚ Φ2 x Φ x . (16) np q“ np´ q

5 In order to prove this we invoke (14) and group the even and odd unitary divisors together. This leads to

˚ 2d µ˚pn{dq d µ˚p2n{dq Φ2 x 1 x 1 x ; np q “ p ´ q p ´ q 2d 2n d n ź|| ź|| 2 ˚ ˚ 1 x d µ pn{dq 1 xd ´µ pn{dq; “ p ´ q p ´ q d n d n ź|| ź|| ˚ 1 xd µ pn{dq Φ˚ x . “ p ` q “ np´ q d n ź|| Let k 1 be an integer and p ∤ n a prime. The unitary divisors of pkn come in two ﬂavors: ě those of the form pkd with d n, and those of the form d n. On grouping these together we || || obtain from (14) that ˚ pk ˚ Φn x Φ k x p q. (17) p np q“ Φ˚ x np q Also, k´1 ˚ ˚ pj Φpkn x Φpn x . (18) p q“ j“0 p q ź To see this we write each of the terms appearing in right hand side as a quotient of two unitaries given by (17). We so obtain a quotient of two unitaries, which equals the left hand side of (18) by (17) again. ˚ ϕ˚pnq ϕ˚pnq´1 Let Φ x x b1x b ˚ . It follows, similar to the classical case, that np q“ ` `¨¨¨` ϕ pnq b1 c˚ 1 µ˚ n for every n N. “´ np q“´ p q P 3 Unitary cyclotomic polynomials as products of cyclotomic polynomials

Recall that κ n p is the square-free kernel of n. p q“ p|n Theorem 2. For any naturalś number n we have

˚ Φ x Φd x . (19) np q“ p q d|n κpdq“źκpnq

Proof. Combination of (9) with (5) yields

µ˚pn{dq µ˚pn{dq ˚ d Φ x x 1 Φδ x . (20) np q“ ´ “ p q d n d n δ d ź|| ´ ¯ ź|| ´ ź| ¯ ˚ e We thus ﬁnd that Φ x Φδ x δ , with np q“ δ|n p q

ś ˚ n eδ µ . (21) “ kδ kδ n ÿ|| ´ ¯

6 The exponents eδ are integers that are to be determined. Given a divisor δ of n, we let d be the smallest multiple of δ that is a block divisor of n. Note that if kδ n, then there is an || integer m such that kδ md. The condition kδ n is in general not equivalent with k n δ, “ || || { however the condition md n is equivalent with m n d. Using these observations and (1) we || || { conclude that ˚ n ˚ n ˚ n n eδ µ µ µ ǫ . (22) “ kδ “ md “ md “ d kδ n md n m n d ÿ|| ´ ¯ ÿ|| ´ ¯ ÿ|| { ´ ¯ ´ ¯ It follows that eδ 0, except when n is the smallest multiple of δ that is a block divisor of n “ (which occurs if and only if κ δ κ n ), in which case eδ 1. p q“ p q “ Remark 3. An alternative form of (19) is

Φ˚ x Φ xd , (23) np q“ κpnqp q d| n źκpnq which is obtained on noting that ˚ d Φn x Φd x Φdκpnq x Φκpnq x , p q“ p q“ p q“ n p q d|n dκpnq|n d| κpnq κpdq“źκpnq ź ź p where in the last step we used repeatedly that Φpn x Φn x if p n. p q“ p q | Remark 4. Theorem 1 says that Φ˚ x is an inclusion-exclusion polynomial associated to the np q prime power factorization of n. A formula of Bachman giving the factorization of an inclusion- exclusion polynomial in cyclotomic polynomials (Theorem 26), then leads to an alternative proof of Theorem 2 (Section 6.1). Remark 5. The convolution deﬁned by f g n f d g n d n N p ˛ qp q“ p q p { q p P q d|n κpdq“ÿκpnq was mentioned by Subbarao [23] and investigated by Thrimurthy [25]. It preserves the multi- plicativity of functions, although it is noncommutative and nonassociative. However, as it is easy to check, for any arithmetic functions f, g and h, f g h f g h , (24) p ˛ q ˛ “ ˛ p ˚ q where is the Dirichlet convolution. See also the review MR0480305 (58 # 478) of [25]. ˚ Our next theorem generalizes Theorem 2. Indeed, Theorem 2 follows from (25) on making n ˚ ˚ the choice g n log Φn x (hence f n log x 1 ) and g n log Φ x . In addition, p q “ p q p q “ p ´ q p q “ np q with this choice (26) yields the identity

˚ µpn{dq Φn x Φ x n N , p q“ dp q p P q d|n κpdq“źκpnq expressing a cyclotomic in terms of unitary cyclotomics. Note that if d n, then κ d κ n | p q “ p q holds iﬀ κ n d iﬀ κ n d d. p q | p { q |

7 Theorem 6. Let g, g˚ : N C be arbitrary functions. Put f n g d . Assume that Ñ p q“ d|n p q ř f n g˚ d n N . p q“ p q p P q d n ÿ|| Then g˚ n g d n N (25) p q“ p q p P q d|n κpdq“ÿκpnq and g n g˚ d µ n d n N . (26) p q“ p q p { q p P q d|n κpdq“ÿκpnq Remark 7. In Theorem 6 the function g is multiplicative if and only if g˚ is multiplicative.

Proof of Theorem 6. By M¨obius inversion we have

g n f d µ n d (27) p q“ p q p { q d n ÿ| and g˚ n f d µ˚ n d . (28) p q“ p q p { q d n ÿ|| These identities show that given g, the function g˚ is uniquely determined and reversely. We have ˚ ˚ g n µ n d g δ g δ eδ , p q“ p { q p q“ p q d n δ d δ n ÿ|| ÿ| ÿ| where eδ is given by (21). The proof of (25) is now easily completed on invoking (22), cf. the proof of Theorem 2. Now we prove identity (26). Put f : g, g : 1 (constant 1 function), h : µ in identity “ “ “ (24). This gives g 1 µ g 1 µ . p ˛ q ˛ “ ˛ p ˚ q Here, g 1 g˚ by (25). Also, 1 µ ǫ, which is a basic property of the classical M¨obius ˛ “ ˚ “ function. Since the function ǫ is the identity for the operation, we conclude that ˛ g˚ µ g, ˛ “ completing the proof.

4 Further properties of unitary cyclotomic polynomials

˚ 4.1 Calculation of Φnp˘1q In this section we determine Φ˚ 1 . For completeness and comparison we mention the np˘ q analogous classical results for Φn 1 . p q

8 Let Λ˚ denote the unitary analog of the von Mangoldt function Λ. It is given by

a log p if n pa is a prime power (a 1); Λ˚ n “ ě (29) p q“ #0 otherwise, and satisﬁes Λ˚ d log n (analogous to the classical identity Λ d log n). d||n p q“ d|n p q“ Lemma 8. Weř have ř

0 if n 1; 0 if n 1; “ e ˚ e “ e Φn 1 p if n p ; and Φ 1 p if n p ; p q“ $ “ np q“ $ “ &’1 otherwise, &’1 otherwise, with p a prime number and e%’ 1. %’ ě In terms of the (unitary) von Mangoldt function this can be reformulated as follows.

˚ Lemma 9. We have Φ1 1 0 and Φ1 1 0. For n 1 we have p q“ p q“ ą Λpnq ˚ Λ˚pnq Φn 1 e and Φ 1 e . p q“ np q“ Proof of Lemma 8. From (5) and (8) we obtain (respectively)

n n x 1 x 1 ˚ ´ Φd x and ´ Φ x . x 1 “ p q x 1 “ dp q d n, d 1 d n, d 1 ´ | źą ´ || źą Thus (respectively) ˚ n Φd 1 and n Φ 1 . (30) “ p q “ dp q d n, d 1 d n, d 1 | źą || źą ˚ By M¨obius inversion the latter identities for all n 1 determine Φm 1 and Φm 1 uniquely for ą p q p q ˚ all m 1. It is thus enough to verify that the formulae claimed for Φm 1 and Φ 1 verify ą p q mp q (30), which is evident.

Remark 10. It is possible to prove Lemma 9 with the (unitary) von Mangoldt function naturally appearing in the proof. To do so one proceeds as in the proof of Lemma 8 and deduces (30) and ˚ concludes that these equations uniquely determine Φm 1 and Φ 1 . It remains then (after p q mp q taking logarithms) to prove the well-known (trivial) identity log n d|n, dą1 Λ d d|n Λ d , ˚ “˚ p q“ p q and likewise in the unitary case, log n 1 Λ d Λ d . “ d||n, dą p q“ d||n přq ř It is not much more diﬃcult to evaluateř Φ˚ 1 . ř np´ q Lemma 11. We have 2 if n 1; ´ “ a ˚ $0 if n 2 ; Φn 1 ’ b “ a b p´ q“ ’p if n 2 p ; &’ “ 1 otherwise, ’ with p an odd prime and a, b 1. ’ ě %

9 Proof. Follows from the identity (23), Lemma 8 and the well-known result

2 if n 1; ´ “ $0 if n 2; Φn 1 ’ “ e (31) p´ q“ ’p if n 2p ; &’ “ 1 otherwise, ’ ’ with p 2 a prime number and e 1. %’ ě ě Assume that Φ˚ 1 1. By (23) it follows that Φ˚ 1 Φ 1 eΦ 1 f , for some np´ q ‰ np´ q “ κpnqp´ q κpnqp q integers e, f 0. The formulas for Φn 1 then show that κ n 2p, with p an odd prime, and a b ě p˘ q ˚ p q| b ab´b b b so n 2 p , a, b 0. In case a, b 1, we have Φ 1 Φ2p 1 Φ2p 1 Φ2p 1 p , “ ě ě np´ q“ p´ q p q “ p´ q “ by, respectively, (23), Lemma 8 and (31). The remaining cases are left to the reader.

˚ Using identity (23) one can likewise immediately evaluate Φ 1 from Φn 1 . np q p q 4.1.1 Some products involving the cos and sin functions It is known that for any n 2, ě n πj Φn 1 sin p q, n “ ϕpnq j“1 2 pj,nźq“1

n πj Φn 1 cos p´ q , (32) n “ 4 ϕpnq{2 j“1 p´ q pj,nźq“1 proved in [7] (for (32) in case n is odd only) and [28] (for any n 2) by two diﬀerent methods, ě see also [13]. Here we provide the unitary analogs of these products, which in combination with the results of the previous section allow one to explicitly evaluate them.

Theorem 12. For any n 2, ě n πj Φ˚ 1 sin np q , (33) n “ ϕ˚pnq j“1 2 pj,nźq˚“1 n πj Φ˚ 1 cos np´ q . (34) n “ 4 ϕ˚pnq{2 j“1 p´ q pj,nźq˚“1 Proof. We adapt the approach in [28] to the unitary case. We need the simple formula

n nϕ˚ n S˚ n : j p q n 2 , (35) p q “ “ 2 p ě q j“1 pj,nÿq˚“1 which can be shown similarly to the usual case. We will only prove (34), the proof of (33) being similar. The product in the left hand side of (34) we denote by P ˚ n . p q

10 If n 2a, a 1, then we note that 2a´1, 2a 1 and so the product in (34) is zero. By “ ě p q˚ “ Lemma 11 it follows that also Φ˚ 1 is zero and thus in this case (34) holds. Therefore we np´ q may assume that n has an odd prime factor, which implies that ϕ˚ n is even. By (7) and (35) p q we then see that

2 2 2 Φ˚ 1 1 ζj ζj{ ζj{ ζ´j{ np´ q “ ´ ´ n “ ´ n n ` n pj,nq “1 pj,nq “1 ź˚ ` ˘ ź˚ ´ ¯ ´ ¯ ˚ 2 ˚ ˚ 2 2ϕ pnqP ˚ n ζj{ 2 ϕ pnqζS pnq{ P ˚ n “ p q ´ n “ p´ q n p q j,n 1 p źq˚“ ´ ¯ ϕ˚pnq nϕ˚pnq{4 ˚ ϕ˚pnq ˚ 2 ζn P n 2i P n , “ p´ q ˚ 2 p q“p´ q p q 4 ϕ pnq{ P ˚ n , “ p´ q p q completing the proof of (34).

Remark 13. A completely similar argument leads to a proof of (32). The argument in that case is even easier, as ϕ n is even for n 3 and it is not necessary to deal with the powers of p q ě two separately.

˚ 4.1.2 Calculation of Φn at other roots of unity

It is known how to explicitly evaluate Φn ζm for m 3, 4, 5, 6, 8, 10, 12 , see [3]. This in p q P t ˚ u combination with identity (23) then allows one to evaluate Φ ζm for these values of m. np q 4.2 Unitary version of Schramm’s identity In this section x will be a real variable. It was proved by Schramm [19] that

n cosp2πj{nq pj,nq Φn x x 1 x 1, n N . p q“ j“1 ´ p ą P q ź ´ ¯ We will prove the following unitary analog.

Theorem 14. We have n cosp2πj{nq Φ˚ x xpj,nq˚ 1 x 1, n N . np q“ ´ p ą P q j“1 ź ´ ¯ This is, in fact, a corollary of a more general identity concerning the discrete Fourier transform (DFT)

n F ˚ m,n : f k,n ζkm (36) f p q “ pp q˚q n k“1 ÿ of functions involving the quantity k,n . p q˚

11 Theorem 15. Let f be an arbitrary arithmetic function. For every m,n 1, ě F ˚ m,n d µ˚ f n d . (37) f p q“ p ˆ qp { q d m,n |pÿ q˚ Furthermore, we have F ˚ m,n f d c˚ m . (38) f p q“ p q n{dp q d n ÿ|| Proof. We have by using that d k,n if and only if d k and d n, || p q˚ | || n F ˚ m,n ζkm µ˚ f d f p q“ n p ˆ qp q k“1 d k,n ÿ ||pÿ q˚ n{d n µ˚ f d ζjm µ˚ f d , “ p ˆ qp q n{d “ p ˆ qp q d d||n j“1 d||n ÿ ÿ pn{ÿdq|m which proves (37). By grouping the terms according to the values of k,n d, we have p q˚ “ r“n{d F ˚ m,n f d ζrm f d c˚ m , f p q“ p q n{d “ p q n{dp q d||n r“1 d|n ÿ pr,n{ÿdq˚“1 ÿ which proves (38),

Remark 16. By (2) and (3) the unitary Ramanujan sum satisﬁes c.˚ m ̺. m µ˚. Thus, p q“ p qˆ the sum in (37) equals ̺. m µ˚ f n c.˚ m f n , leading to another proof of (38). p p qˆ ˆ qp q “ p p qˆ qp q Remark 17. Identity (37) shows that if f is a real valued function, then so is F ˚ m,n and f p q hence in this case the factor ζkm in (36) can be replaced by cos 2πkm n . More exactly, if f is n p { q a real valued function, then

n f k,n cos 2πkm n d µ˚ f n d , (39) pp q˚q p { q“ p ˆ qp { q k“1 d m,n ÿ |pÿ q˚ n f k,n sin 2πkm n 0. pp q˚q p { q“ k“1 ÿ In the special case f n n (n N) and m 1 we obtain the following identities: p q“ P “ n µ˚ d n k,n cos 2πk n n p q ϕ˚ n and k,n sin 2πk n 0. p q˚ p { q“ d “ p q p q˚ p { q“ k“1 d n k“1 ÿ ÿ|| ÿ In the classical case where k,n is replaced by k,n and ϕ˚ n is replaced by ϕ n , these were p q˚ p q p q p q pointed out by Schramm [18, 19].

12 Proof of Theorem 14. By taking f n log xn 1 we have by (9) that p q“ p ´ q µ˚ f n µ˚ d log xn{d 1 log Φ˚ x . p ˆ qp q“ p q p ´ q“ np q d n ÿ|| The assumption that x 1 ensures that f is real. It then follows from (39) that ą n cosp2πjm{nq xpj,nq˚ 1 Φ˚ x d m,n N . ´ “ n{dp q p P q j“1 d m,n ź ´ ¯ ||pź q˚ The proof is completed on putting m 1. “ 5 The coeﬃcients of unitary cyclotomic polynomials

We write 8 Φ˚ x a˚ j xj . (40) np q“ np q j“0 ÿ This notation looks perhaps strange to the reader, but implicitly defines the coefficients for every j, which serves our purposes. In [11] the following result is proven. Theorem 18. Let m 1 be fixed. We have a˚ j : n 1, j 0 Z. ě t mnp q ě ě u“ Given any polynomial f, its height h f is defined as its maximum coefficient in absolute p q value. Conjecture 19. For any given natural number m there is a cyclotomic polynomial having height m. This conjecture was put forward by Kosyak et al. [12]. Here we propose the following conjecture. Conjecture 20. For any given natural number m there is a unitary cyclotomic polynomial having height m. These two conjectures are closely connected. Proposition 21. If Conjecture 19 is true, then so is Conjecture 20.

Proof. Suppose that h Φn m. Then, by elementary properties of cyclotomic polynomials, p q “ h Φ m. Now note that h Φ˚ h Φ (since Φ˚ Φ ). p κpnqq“ p κpnqq“ p κpnqq κpnq “ κpnq The best result available to date regarding these two conjectures is the following. Theorem 22. Almost all positive integers occur as the height of an (unitary) cyclotomic polynomial. Specifically, for any fixed ǫ 0, the number of positive integers x that do not occur 3{5`ǫ ą ď 1{2`ǫ as a height is ǫ x . Under the Lindelöf Hypothesis this number is ǫ x . ! ! Proof. The result is actually a corollary of [12, Theorem 4]. In that theorem only certain special ˚ cyclotomic polynomials of the form Φpqr, with p q r primes, feature. As Φ Φpqr, we ă ă pqr “ are done.

13 This theorem is deep, as it relies on deep results from analytic number theory on gaps between consecutive primes. Let k be a squarefree integer. Consider the set

B k : h Φ˚ : κ n k . (41) p q “ t p nq p q“ u ˚ Note that if we replace h Φ by h Φn this set will be h Φk . p nq p q t p qu Lemma 23. Let k be a squarefree integer. Suppose that k has at most two distinct prime factors. Then B k 1 . p q “ t u Proof. If k is a prime power, the conclusion follows from (13). If k has precisely two distinct ˚ prime factors, say p and q, we can write Φ x Q e f x by Theorem 1. The polynomial np q “ tp ,q up q Q e f x can be interpreted as the semigroup polynomial associated to the numerical semi- tp ,q up q group pe,qf and as such will have height 1 (see Jones et al. [11] or Moree [16]). A shorter, but x y less conceptual, proof is obtained on merely invoking Lemma 4 of [8].

˚ Remark 24. By the above proof and (16) we also have h Φ2peqf 1. However, it is not always ˚ ˚ p q“ true that h Φ 1 (for example h Φ60 2). p 4peqf q“ p q“ Computer work by Bin Zhang suggests that B k will be large if k has at least three prime p q factors. This suggests that perhaps there is hope of proving a stronger result on heights assumed by unitary cyclotomic polynomials than is provided by Theorem 22.

Question 25. Suppose that k has at least three odd prime factors. Is B k unbounded? p q If k has four or more prime factors, we would not be surprised if B k is unbounded. If it p q has precisely three factors, the situation is not so clear. For example,

7 max h Φ˚ : n 2a3b5c, a 0, b 0, c 0, 1 n 10 15. t p nq “ ą ą ą ă ă u“ 6 More on inclusion-exclusion polynomials

The following result gives the factorization of an inclusion-exclusion polynomial Qρ x into p q cyclotomics.

Theorem 26 (Bachman, [2]). Let ρ r1, r2, . . . , rs be a set of increasing natural numbers “ t u satisfying ri 1 and ri, rj 1 for i j. Put ą p q“ ‰ s Dρ d : d ri and d, ri 1 for all i . “ t | i“1 p qą u ź Then Qρ x Φd x . p q“ dPDρ p q Our proof ofś Theorem 1 makes use of the following basic property of inclusion-exclusion polynomials.

Theorem 27. Suppose that Qρ1 x Qρ2 x , then ρ1 ρ2. p q“ p q “

14 Its proof rests on the following easy lemma. Lemma 28. A Kronecker polynomial can be written as

f x xs xd 1 ed , (42) p q“ p ´ q dPD ź where D is a unique ﬁnite set of integers and s 0 and the ed 0 are unique integers. ě ‰ First proof. From (11) and (6) we deduce that f x can be written as claimed. It remains p q to show the uniqueness. The integer s 0 is merely the order of f x in x 0. We start ě p q “ by taking D to be the empty set. We now consider f x xs around x 0. Either f x 1 p q{ “ d1 d1 p1 q “ and we are done, or for some ǫ1 1, 1 we have ǫ1f x 1 ax O x ` with P t´ u p q“ ´ ` ` p1 q a 0 an integer. In case a is odd we have xd1 1 a 1 axd1 O xd1` , in case a ‰ d1 a d1 pd1`1 ´ q “ ´ ` D ` p q is even we have x 1 1 ax O x . We add d1 to and put ed1 a. Put d1p ´ a q “ ´ ` p q “ g x : f x x 1 ´ . Note that either g x 1 or for an appropriate ǫ2 1, 1 the p q “ p qp ´ q d2 d2 1 p q “ P t´ u identity ǫ2g x 1 bx O x ` , with d2 d1 and b 0 holds. We now add d2 to D and p q“´ ` ` p q ą ‰ put ed2 b. We continue in this way until we arrive at the polynomial 1 (and this will happen “ as we know a priori that D is ﬁnite).

s e1 e Second proof. By (11) we can write f x x x 1 g x , with g x 2 Φd x d . As g x p q“ p ´ q p q p q“ dě p q p q is the product of cyclotomic Φd with d 2, it is selfreciprocal and satisfies g 0 1. Note ě ś p q “ that s and e1 are the order of f x in x 0, respectively x 1, and therefore unique. Thus, p q “ “ w.l.o.g., we may assume that f x is selfreciprocal and satisfies f 0 1. Around x 0 we p q d1 d1 1 p q “ “ either have f x 1 or f x 1 ax O x ` with a 0 an integer and d1 1. Note that pd1 q“a d1p 1q“ ´ ` p q ‰ ě f x 1 x O x ` . We start with setting D to be the empty set. We add d1 to D and p q “ p ´ q ` p q d1 á d2 d2`1 put ed1 a. Put g x : f x 1 x . Note that either g x 1 or g x 1 bx O x “ p q “ p qp ´ q D p q“ p q“ ´ ` p q with d2 d1 and b 0. We now add d2 to and put ed2 b. We continue in this way until ą ‰ “ we arrive at the polynomial 1 (and this will happen as we know a priori that D is finite). We obtain f x 1 xd ed xd 1 ed , p q“ p ´ q “ p ´ q dPD dPD ź ź with D and the ed unique. Remark 29. By a similar reasoning one can show that if f x Z x satisfies f x 1 mod x , p q P rr ss p q” p q there exist unique integers e1, e2,... such that in Z x rr ss 8 f x 1 xn en . p q“ n“1p ´ q ź This is the so-called Witt expansion which arises in many areas of mathematics, see for example Moree [15] for more information. Proof of Theorem 27. Given an inclusion-exclusion polynomial f x in the standard form f x i p q p q“ aix , we give a procedure leading to a unique ρ r1, . . . , rm such that f x Qρ x . As i “ t u p q “ p q f x is a Kronecker polynomial satisfying f 0 1, by Lemma 28 we can write řp q p q“ f x xd 1 ed , p q“ p ´ q dPD ź 15 where D and the ed 0 are unique and computable. Comparison with (10) then shows that ‰ D n0,ni,nij,nijk,... . The question is know which ρ correspond to the set D. We know for “ t u example, that whatever ρ is, the product of its ri is certainly unique and equals n0 max D. “ The numbers ri themselves, also turn out to be unique. We order the elements in D in such a way that d1 d2 .... We put r1 d1. If d2 is coprime with r1 we put r2 d2, if not we ă ă “ “ consider d3. If an di is coprime to every r1, . . . , rg we have at a certain point, we put rg 1 di. ` “ If rm is the last r number so found, we have f x Qρ x with ρ r1, . . . , rm uniquely p q “ p q “ t u determined.

6.1 Unitary cyclotomics as inclusion-exclusion polynomials Armed with Theorem 27 we are now ready to prove Theorem 1.

s ei e1 Proof of Theorem 1. Let n 2 be an integer and i“1 pi its canonical factorization with p1 e2 es ě ˚ ă p . . . p . On comparing formula (9) with (10) it follows that Φ x Q e1 es x . Re- 2 s n tp1 ,...,ps u ă ă ś e1 es p q“ p q versely, given any ascending sequence of prime powers p1 . . . ps with distinct base primes ă ă ˚ s ei p1,...,ps, the polynomial Q e1 es x is seen to correspond to Φ x with n 1 p . The tp1 ,..., ps up q np q “ i“ i one-to-one part of the claim is a consequence of Theorem 27. ś Theorem 1 together with Theorem 26 can be used to reprove Theorem 2:

Alternative proof of Theorem 2. For n 1 the result is obviously true. Now let n 2 be s ei “ e1 e2 es ě an integer and i“1 pi its canonical factorization with p1 p2 . . . ps . Put ρ e1 es ă ă ă “ p1 ,...,p . Then by Theorem 1, respectively Theorem 26, t s u ś ˚ Φ x Qρ x Φd x , np q“ p q“ p q dPD źρ with Dρ d : d n, p1 ps d d : d n, k d k n . “ t | ¨ ¨ ¨ | u “ t | p q“ p qu 7 Applications of Theorem 6

˚ ˚ By selecting g n cn k , g n c k (Ramanujan, resp. unitary Ramanujan sums), p q “ p q p q “ np q where f n ̺n k , we deduce from Theorem 6 the following identities. p q“ p q Corollary 30. For any n, k N we have P ˚ c k cd k (43) np q“ p q d|n κpdq“ÿκpnq and ˚ cn k c k µ n d . p q“ d p q p { q d|n κpdq“ÿκpnq

s Let ids n n (s R). In the case g n Js n : µ ids n (Jordan function of order ˚ p q“˚ P˚ p q “ p q “ p ˚ qp q s s), g n J n : µ ids n , the unitary Jordan function of order s, where f n n , we p q“ s p q “ p ˆ qp q p q“ deduce the following corollary of Theorem 6.

16 Corollary 31. For every n N, s R we have P P ˚ J n Js d (44) s p q“ p q d|n κpdq“ÿκpnq and ˚ Js n J d µ n d . p q“ s p q p { q d|n κpdq“ÿκpnq

Identity (43) was deduced by McCarthy [14, Ch. 4], while (44) was obtained by Cohen [5, ˚ ˚ Lemma 3.1] using diﬀerent reasonings. If s 1, then J1 n ϕ n , J1 n ϕ n and we “ p q “ p q p q “ p q deduce the following corollary.

Corollary 32. We have ϕ˚ n ϕ d n N . p q“ p q p P q d|n κpdq“ÿκpnq This result also follows by setting k n in Corollary 30, or by comparing the degrees of the “ polynomials in (19). We remark that Cohen [5, Lemma 4.1] also showed that the identity

κs n dsµ2 d n N p q“ p q p P q d|n κpdq“ÿκpnq holds for any s. Putting k 1 in Corollary 30 and noting that κ n is the only squarefree divisor d of n “ p q satisfying κ d κ n , we obtain the following corollary. p q“ p q Corollary 33. We have

µ˚ n µ d µ κ n n N . p q“ p q“ p p qq p P q d|n κpdq“ÿκpnq

On taking g n Λ n , g˚ n Λ˚ n and by the well-known identity Λ d log n, p q “ p q p q “ p q d|n p q “ we see that f n log n and obtain the ﬁnal corollary. p q“ ř Corollary 34. We have Λ˚ n Λ d n N . p q“ p q p P q d|n κpdq“ÿκpnq

From it we deduce the truth of (29).

17 8 Connection with unitary Ramanujan sums

Certain formulas concerning the unitary cyclotomic polynomials and unitary Ramanujan sums can be easily deduced from their classical analogues, by using the above identities. For example, we have Corollary 35. For any n 1 and x C, x 1, ą P | |ă 8 c˚ k Φ˚ x exp np qxk . (45) np q“ ´ k ˜ k“1 ¸ ÿ Proof. It is known that for any n 1 and x 1, ą | |ă 8 cn k k Φn x exp p qx , (46) p q“ ´ k ˜ k“1 ¸ ÿ see Nicol [17, Cor. 3.2], T´oth [27, Th. 1] or Herrera-Poyatos and Moree [9]. By Theorem 2 and the identity (43) we then obtain

8 k 8 ˚ ˚ x cn k k Φ x Φd x exp ¨ cd k ˛ exp p qx , np q“ p q“ ´ k p q “ ´ k d|n k“1 d|n ˜ k“1 ¸ ź ˚ ÿ ÿ ‹ ÿ κpdq“κpnq ˚ κpdq“κpnq ‹ ˝ ‚ completing the proof.

8 For any n 1 the series 1 cn k k converges, see, e.g., Hölder [10]. Therefore, Lemma 9 ą k“ p q{ in combination with (46) gives ř 8 cn k log Φn 1 Λ n p q n 1 . (47) p q“ p q“´ k p ą q k“1 ÿ The second identity was first discovered by Ramanujan and expresses an arithmetic function as an infinite series involving Ramanujan sums (for a different proof see Sivaramakrishnan [21, Theorem 87]). Such expressions are now called Ramanujan expansions, see, e.g., Schwarz and Spilker [20, Chapter VIII]. 8 The convergence of k“1 cn k k for n 1 in combination with (43) shows that also 8 ˚ p q{ ą 1 c k k converges for any n 1. Thus, by Lemma 9 again, k“ np q{ ř ą ř 8 c˚ k log Φ˚ 1 Λ˚ n np q n 1 . (48) np q“ p q“´ k p ą q k“1 ÿ The second identity in (48) was obtained by Subbarao [22], without referring to unitary cyclotomic polynomials and with an incomplete proof, namely without showing that the corresponding series converges. Acknowledgments. Part of this work was done when the second author visited the Max Planck Institute of Mathematics in Bonn, Germany, in February 2017. He thanks the Institute for the invitation and support. We thank Gennady Bachman for correspondence that led to a simplification of the proof of Theorem 27. Bin Zhang (Qufu Normal University), kindly did computer work on B k (defined in (41)) on request of the first author. p q

18 References

[1] T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathemat- ics, Springer-Verlag, New York-Heidelberg, 1976.

[2] G. Bachman, On ternary inclusion-exclusion polynomials, Integers 10 (2010), Paper A48, 623–638.

[3] B. Bzd¸ega, A. Herrera-Poyatos and P. Moree, Cyclotomic polynomials at roots of unity, Acta Arith. 184 (2018), 215–230.

[4] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66–80.

[5] E. Cohen, An elementary method in the asymptotic theory of numbers, Duke. Math. J. 28 (1961), 183–192.

[6] P. A. Damianou, Monic polynomials in Z x with roots in the unit disc, Amer. Math. r s Monthly 108 (2001), 253–257.

[7] M. Deaconescu and I. S´andor, Variations on a theme by Hurwitz, Gaz. Mat., Perfect¸. Metod. Metodol. Mat. Inf. (Bucharest) 8 (1987), No. 4, 186–191.

[8] A. Decker and P. Moree, Coeﬃcient convexity of divisors of xn 1, Sarajevo J. Math. 9 (21) ´ (2013), 3–28.

[9] A. Herrera-Poyatos and P. Moree, Coeﬃcients and higher order derivatives of cyclotomic polynomials: old and new, https://arxiv.org/abs/1805.05207, Expos. Math., to appear.

[10] O. H¨older, Zur Theorie der Kreisteilungsgleichung Km x 0, Prace Mat.-Fiz. 43 (1936), p q “ 13–23.

[11] G. Jones, P. Kester, L. Martirosyan, P. Moree, L. T´oth, B. White and B. Zhang, Coeﬃcients of (inverse) unitary cyclotomic polynomials, preprint MPIM2019-35, Kodai Math. J., to appear.

[12] A. Kosyak, P. Moree, E. Sofos and B. Zhang, Cyclotomic polynomials with prescribed height and prime number theory, https://arxiv.org/abs/1910.01039, submitted for publica- tion.

[13] P. Luschny and S. Wehmeier, The lcm 1, 2,...,n as a product of sine values sampled over p q the points in Farey sequences, Preprint, 2009, https://arxiv.org/abs/0909.1838, 4 pp.

[14] P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, 1986.

[15] P. Moree, Approximation of singular series and automata. With an appendix by Gerhard Niklasch, Manuscripta Math. 101 (2000), 385–399.

[16] P. Moree, Numerical semigroups, cyclotomic polynomials, and Bernoulli numbers, Amer. Math. Monthly 121 (2014), 890–902.

19 [17] C. A. Nicol, Some formulas involving Ramanujan sums, Canad. J. Math. 14 (1962), 284– 286.

[18] W. Schramm, The Fourier transform of functions of the greatest common divisor, Integers 8 (2008), Paper A50, 7 pp.

[19] W. Schramm, Eine alternative Produktdarstellung f¨ur die Kreisteilungspolynome, Elem. Math. 70 (2015), 137–143.

[20] W. Schwarz and J. Spilker, Arithmetical functions, London Mathematical Society Lecture Note Series 184, Cambridge University Press, Cambridge, 1994.

[21] R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Monographs and Text- books in Pure and Applied Mathematics, Vol. 126, Marcel Dekker, 1989.

[22] M. V. Subbarao, A note on the arithmetic functions C n, r and C˚ n, r , Nieuw Arch. p q p q Wisk. (3) 14 (1966) 237–240.

[23] M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251, Springer, 1972, 247–271.

[24] R. Thangadurai, On the coeﬃcients of cyclotomic polynomials, Cyclotomic ﬁelds and related topics (Pune, 1999), 311–322.

[25] P. Thrimurthy, On certain arithmetic convolutions, Vidya B 20 (1977), 72–78.

[26] L. T´oth, On the bi-unitary analogues of Euler’s arithmetical function and the gcd-sum function, J. Integer Seq. 12 (2009), Article 9.5.2, 10 pp.

[27] L. T´oth, Some remarks on Ramanujan sums and cyclotomic polynomials, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), 277–292.

[28] M. T¸ena, Roots of unity (in Romanian), Bibl. Soc. S¸t. Mat. Romania, 2005, 164 pp.

[29] R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc., 33 (1931), 579–662.

[30] S. H. Weintraub, Several proofs of the irreducibility of the cyclotomic polynomials, Amer. Math. Monthly 120 (2013), 537–545.

Pieter Moree Max-Planck-Institut für Mathematik Vivatsgasse 7, 53111 Bonn, Germany E-mail: [email protected] LászlóTóth Department of Mathematics University of Pécs Ifjúság útja 6, 7624 Pécs, Hungary E-mail: [email protected]