#A63 INTEGERS 21 (2021) NEWTON SEQUENCES AND DIRICHLET CONVOLUTION Klaudiusz W´ojcik Faculty of Mathematics, Jagiellonian University, Krak´ow,Poland [email protected] Received: 12/4/20, Revised: 4/20/21, Accepted: 5/29/21, Published: 6/3/21 Abstract An integer sequence a : N −! Z is called a Newton sequence generated by the sequence of integers c : N −! Z, if the following Newton identities hold a(n) = c(1) a(n − 1) + ::: + c(n − 1) a(1) + n c(n): We show that a is a Newton sequence if and only if X f(d) a(n=d) ≡ 0 (mod n); n ≥ 1; djn for every function f : N −! Z satisfying the two conditions X f(d) ≡ 0 (mod n); f(1) = ±1: djn In particular, f may be the M¨obiusfunction, the Euler totient function or the Jordan totient function. 1. Introduction We recall some basic facts concerning Dirichlet involution (see [2]). Let A(C) be the set of all complex-valued arithmetical functions, i.e., the set of functions defined on N with values in C. By A(Z) ⊂ A(C) we denote the set of all integer-valued sequences. The Dirichlet convolution of functions f; g 2 A(C) is the function f ∗ g 2 A(C) defined by X X (f ∗ g)(n) = f(d)g(n=d) = f(d1)g(d2); n ≥ 1: djn d1d2=n For any non-empty subsets A; B of A(C) we put A ∗ B := fa ∗ b : a 2 A; b 2 Bg: We distinguish the following integer-valued sequences: INTEGERS: 21 (2021) 2 () the multiplicative identity with respect to Dirichlet convolution: ( 1 1; if n = 1 (n) = = n 0; if n > 1; (µ) the M¨obiusfunction µ: 8 1; if n = 1 <> µ(n) = (−1)r; if n is a product of r different primes :>0; otherwise: k (Ik) Ik(n) = n for k ≥ 0, (Sf ) the divisor sum over f 2 A(Z): X Sf (n) := f ∗ I0(n) = f(d); n ≥ 1; djn (') the Euler totient function ': X '(n) = µ ∗ I1(n) = 1; 1≤k≤n (k;n)=1 (Jk) the Jordan totient function Jk: X Jk(n) = µ ∗ Ik(n) = 1: 1≤a1;:::;ak≤n (a1;:::;ak;n)=1 The set of arithmetical functions A(C) is a commutative ring under pointwise ad- dition and Dirichlet convolution. The invertible elements of this ring are the arith- metical functions f with f(1) 6= 0. The following well-known facts will be useful later ([2]): (1) Sµ = µ ∗ I0 = , (2) the M¨obiusinversion formula: Sf = f ∗ I0 if and only if f = µ ∗ Sf , (3) S' = I1, (4) SJk = Ik. Definition 1. Let a 2 A(Z) be a sequence of integers. We say that a ≡ 0, if a(n) ≡ 0 (mod n); for all n ≥ 1. INTEGERS: 21 (2021) 3 We put ∗ A (Z) := a 2 A(Z): a(1) = ±1 ; A0(Z) := a 2 A(Z): a ≡ 0 ; ∗ ∗ A0(Z) :=A (Z) \A0(Z); ∗ M(Z) := f 2 A(Z): Sf 2 A0(Z) : ∗ Example 2. It follows that Ik 2 A0(Z) and µ, '; Jk 2 M(Z). Corollary 3. (1) Every a 2 A∗(Z) admits an inverse a−1 2 A(Z), (2) A0(Z) ∗ A0(Z) ⊂ A0(Z), ∗ ∗ ∗ (3) A0(Z) ∗ A0(Z) ⊂ A0(Z), ∗ (4) (A0(Z); ∗) is the commutative group, ∗ (5) M(Z) = µ ∗ g : g 2 A0(Z) . Proof. The conditions (1) and (5) are obvious. If a; b 2 A0(Z), then X a ∗ b(n) = a(d)b(n=d) ≡ 0 (mod n); djn −1 ∗ hence (2) follows. We claim that a ≡ 0 for a 2 A0(Z). We use the induction on n. It is trivial for n = 1. For n > 1, by [[2], Theorem 2.8] and the inductive step, we get that X a−1(n) = ± a(d)a−1(n=d) ≡ 0 (mod n): djn d>1 Definition 4 (Newton sequence). A sequence of integers a 2 A(Z) is called a Newton sequence generated by the sequence of integers c 2 A(Z), if the following Newton identities hold: for all n 2 N a(n) = c(1) a(n − 1) + ::: + c(n − 1) a(1) + n c(n): Denote by AN (Z) the set of Newton sequences, i.e., AN (Z) = fa : a is a Newton sequence generated by a sequence of integers cg: INTEGERS: 21 (2021) 4 Example 5 (Sequence of traces). For a square integer matrix A 2 Ml(Z) we define a sequence of traces tr [A] 2 A(Z) by n (tr [A])n := tr A ; n ≥ 1: It follows by the classical Newton's identities that a Newton sequence a 2 A(Z) is generated by a finite sequence c = (c(1); : : : ; c(l)) 2 A(Z) (i.e., c(n) = 0 for n > l) if and only if a is the sequence of traces tr [A], where A is the companion matrix of the monic polynomial w(x) = xl − c(1)xl−1 − ::: − c(l − 1)x − c(l) 2 Z[x], i.e., 2 0 0 ::: 0 c(l) 3 6 1 0 ::: 0 c(l − 1) 7 6 7 6 0 1 ::: 0 c(l − 2) 7 A := A[c(1); : : : ; c(l)] := 6 7 2 Ml(Z): 6 . .. 7 4 . 0 . 0 . 5 0 ::: 0 1 c(1) Since w(X) = det (XIl − A) 2 Z[X], the eigenvalues of A are the roots of w. Definition 6 (Dold-Fermat sequence [6]). A sequence a 2 A(Z) is called a Dold- Fermat sequence, if µ ∗ a 2 A0(Z), i.e., X µ(d) a(n=d) ≡ 0 (mod n); n ≥ 1: djn In combinatorics and number theory, Dold-Fermat sequences have also been called pre-realizable sequences, relatively realizable sequences, Gauss sequences and generalized Fermat sequences ([1, 3, 7, 8, 9, 10, 11, 13, 14]). We prove the following theorem. Theorem 7. For a 2 A(Z) the following conditions are equivalent: (1) a is a Newton sequence, (2) a is a Dold-Fermat sequence, (3) f ∗ a 2 A0(Z), i.e., X f(d) a(n=d) ≡ 0 (mod n); n ≥ 1; djn for every f 2 M(Z). The equivalence of the conditions (1) and (2) was proved in [8]. On the other hand, it was observed in [1] that in the definition of the Dold-Fermat sequence the M¨obiusfunction can be replaced by the Euler totient function '. In the last section, we define the unitary analog of the Newton sequences. In particular, we show that a unitary Newton sequence is periodic if and only if it is bounded. We also prove that a p-periodic unitary sequence with prime period has to be constant. INTEGERS: 21 (2021) 5 2. Newton Sequences Lemma 8. Let a 2 A(Z) and f 2 M(Z). The following conditions are equivalent: (1) µ ∗ a 2 A0(Z), (2) f ∗ a 2 A0(Z). Proof. We first show that (1) implies (2). We have f ∗ a = ∗ f ∗ a = µ ∗ I0 ∗ f ∗ a = (µ ∗ a) ∗ Sf 2 A0(Z); by Corollary 3. Assume that (2) holds. We have (µ ∗ a) ∗ Sf = f ∗ a 2 A0(Z): ∗ −1 Since Sf belongs to A0(Z), Sf does and by Corollary 3, one has −1 µ ∗ a = Sf ∗ (f ∗ a) 2 A0(Z): Remark 9. Observe that in the proof of implication (1) =) (2) we need only that Sf 2 A0(Z) and f does not have to be invertible. In particular, one can take f of the form f = µ ∗ g for some g 2 A0(Z). As a consequence, one has the following result. Theorem 10. Assume that a 2 A(Z). The following conditions are equivalent: (1) a 2 AN (Z), (2) µ ∗ a 2 A0(Z), (3) f ∗ a 2 A0(Z) for every f 2 M(Z). Proof. The equivalence of the conditions (1) and (2) was proved in [8]. Lemma 11. Let f 2 M(Z). Then −1 F : A0(Z) 3 b 7−! f ∗ b 2 AN (Z) is a bijective map. −1 −1 Proof. If b 2 A0(Z) then f ∗ f ∗ b = b 2 A0(Z), hence f ∗ b 2 AN (Z) by Theorem 10, so F is well-defined. Since F is obviously injective, it is sufficient to show its surjectivity. If a 2 AN (Z) then f ∗ a 2 A0(Z), hence f ∗ a = b for some −1 b 2 A0(Z), so a = f ∗ b. INTEGERS: 21 (2021) 6 Corollary 12. The following maps are bijective: −1 (a) A0(Z) 3 b 7−! µ ∗ b = Sb 2 AN (Z), −1 (b) A0(Z) 3 b 7−! ' ∗ b = Sµ·I1 ∗ b 2 AN (Z). Example 13 (Newton sequences). The following well-known sequences can be interpreted as Newton sequences: P (1) σ1(n) = SI1 (n) = djn d; P k (2) σk(n) = SIk (n) = djn d ; P (3) Sµ·I1 ∗ (n!) = djn dµ(d)(n=d)!; P k P nk (4) Sµ·I1 ∗ Ik(n) = djn dµ(d)(n=d) = djn µ(d) dk−1 : Lemma 14. If a 2 AN (Z) and b 2 A0(Z), then a ∗ b 2 AN (Z). Proof. Since b; µ ∗ a 2 A0(Z), we have µ ∗ (a ∗ b) = b ∗ (µ ∗ a) 2 A0(Z); so a ∗ b 2 AN (Z). Example 15 (Newton sequences). The following sequences are Newton sequences: P k (1) a ∗ Ik(n) = djn d a(n=d); P (2) b(n) = djn a(n=d) d!: Corollary 16.