Thai Journal of Mathematics : (2020) 127{134 Special Issue: The 14th IMT-GT ICMSA 2018 http://thaijmath.in.cmu.ac.th Online ISSN 1686-0209 Unitary Analogues of Some Arithmetic Functions Pussadee Yangklany;1, Vichian Laohakosolz, Pattira Ruengsinsubz and Sukrawan Mavechay yDepartment of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand e-mail : pussadee [email protected] and [email protected] zDepartment of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand e-mail : [email protected] and [email protected] Abstract : A number of arithmetic functions, referred to as the generalized unitary-Euler's totient, generalized unitary-Cohen's totient, generalized unitary-divisor, generalized unitary-Liouville, odd-phi, and even-phi functions which generalize the classical totient, divisor and Liouville functions, are intro- duced in the setting of unitary convolution. Basic properties of these functions extending the existing ones are established. Results related to the problem of counting exponentially odd and exponentially even numbers are derived as applications. Keywords : arithmetic function; unitary convolution; multiplicative function 2010 Mathematics Subject Classification : 11A25 (2000 MSC) 1 Introduction An arithmetic function, [1], is a complex-valued function defined on the set of positive integers. The set of arithmetic functions, A, with addition +, and unitary convolution t defined, respectively, by X (f + g)(n) = f(n) + g(n); (f t g)(n) = f (n=d) g(d); djjn where djjn denotes the unitary divisor (i.e., those divisors d of n for which gcd(d; n=d) = 1), is a commu- tative ring with zero divisors [2]. The identity element under the unitary convolution is the function ( 1 if n = 1 I(n) = 0 if n > 1: A non-zero arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n) whenever gcd(m; n) = 1: Note that a unitary convolution of two multiplicative functions is a multiplicative function. 1Corresponding author. Copyright c 2020 by the Mathematical Association of Thailand. All rights reserved. 128 Thai J. M ath. Special Issue (2020)/ P. Yangklan, V. Laohakosol, P. Ruengsinsub and S. Mavecha A modern systematic study of unitary convolution seems to be started by Cohen. In [3], mimicking the classical case, he introduced the unitary Euler's totient as follows: for a; b 2 Z with b > 0, denote by (a; b)∗ the greatest divisor of a which is a unitary divisor of b; when (a; b)∗ = 1, the integer a is said to be semiprime to b. The unitary Euler's totient, ', is defined to be the number of positive integers that are semiprime to n, i.e., X '(n) = 1: x≤n (x;n)∗=1 A semi-reduced residue system mod n, denoted by SRRS(n), is a set of integers in a residue system mod n that are semiprime to n. In the same paper, Cohen introduced a unitary Ramanujan sum c∗(m; n) by X c∗(m; n) = exp(2πimx=n) x2SRRS(n) where the summation extends over the integers x in a semi-reduced residue system mod n. Based upon this notion, we observe that '(n) = c∗(0; n): Cohen also defined µ(n) = c∗(1; n) and proved a unitary analogue of the M¨obiusinversion formula which states that for f; g 2 A, we have X X f(n) = g(d) () g(n) = µ(d)f(n=d); djjn djjn as well as several identities relating to these functions such as X X !(n) '(d) = n; µ(d) = I(n); '(n) = (ζ1 t µ)(n); µ(n) = (−1) (1.1) djjn djjn where ζ1(n) := n, and !(n) denotes the number of distinct prime factors of n with !(1) = 0. In addition, Cohen introduced the concept of an exponentially odd number which is an integer n 2 N whose prime a1 as factorization takes the form n = p1 : : : ps with all powers ai being odd positive integers. Denote by Eo the set of all exponentially odd numbers. Observe that n 2 Eo whenever its greatest unitary square divisor (the largest unitary divisor which is a square) is 1, and so 1 2 Eo (because its greatest unitary square divisor is 1). Apart from Cohen's work, Rao in [4] gave the following extension of Cohen's totient. For n; m; k 2 N, k ∗ k let (n; m )k denote the largest unitary divisor of m that divides n and is a kth power. Denote the unitary ∗ k k ∗ analogue of Cohen's totient, 'k(m), as the number of positive integers n ≤ m such that (n; m )k = 1. Rao proved that ∗ X k k Y 1 'k(m) = d µ(m=d) = (ζk t µ)(m) = m 1 − ; (1.2) pkνp(m) djjm pjn k where ζk(n) := n and νp(m) denotes the highest power of the prime p that divides m. This result implies ∗ ∗ at once that 'k is a multiplicative function and obviously that '1 = ': Let d(n) denote the number of unitary divisors of n, and let σk(n) denote the sum of the kth power of the unitary divisors of n (see also [5]), i.e., X !(n) X k d(n) = 1 = 2 ; σk(n) = d = (ζk t U)(n): djjn djjn where U(n) := 1 for all n 2 N. Rao also established the identities X ∗ X ∗ k σk(n) = 'k(n=d)d(d); σs+k(d)'k(n=d) = n σs(n): djjn djjn Unitary Analogues of Some Arithmetic Functions 129 Recently, in [6], a generalized unitary M¨obiusfunction was introduced via !(n) µα(n) = (−α) (n 2 N; α 2 C n f0g); µ0 := I; µ1 := µ, µ−1 = U: The main objectives in [6] were (i) to prove the following generalized unitary M¨obiusinversion formula: for f; g 2 A and α 2 C n f0g, we have X n X n f(n) = g(d)µ () g(n) = f µ (d); or symbolically; f = g t µ () g = f t µ ; −α d d α −α α djjn djjn (ii) to derive some characterizations of multiplicative function using generalized unitary M¨obiusfunction. Here, we continue our unitary investigation by further generalizing the unitary totient, unitary divisor and unitary Liouville functions. This is done by introducing the so-called gu-Euler's totient, gu-Cohen's totient, gu-divisor, gu-Liouville, odd-phi, and even-phi functions (the abbreviation gu stands for gener- alized unitary). Apart from deriving properties of these functions which extend the existing ones, in the last section, we apply these results to functions that are related to the problem of counting exponentially odd and exponentially even numbers. 2 Preliminaries There are two parts in this section; generalized unitary analogues are defined in the first part and their properties are established in the second part. 2.1 Definitions Let α 2 R. 1. The gu-Euler's totient, 'α, is defined by 'α = ζ1 t µα: ∗ 2. The gu-Cohen's totient, 'α, is defined by ∗ 'α = ζα t µ. 3. The gu-divisor function, σα(n), is defined to be the sum of the αth power of positive unitary divisor d of n, i.e., α X α σ (n) = d (n 2 N): djjn 4. The gu-Liouville function, λα, is defined by Ω(n) λα(n) = (−α) (n 2 N); λ0 := I; where Ω(n) is the number of prime factors of n counting with multiplicity with Ω(1) = 0. The following facts are easily checked. • The gu-Euler's totient, gu-Cohen's totient, gu-divisor function and gu-Liouville function are mul- tiplicative functions. (P !(n=d) djjn d(−α) if α 2 R n f0g • 'α(n) = ζ1(n) if α = 0: ∗ • For α 2 N, the function 'α is identical with the unitary analogue of the Cohen's totient. α • σ = ζα t µ−1: • σ1 = σ and σ0(n) = d(n). Ω(n) • λ1 is identical with the classical Liouville function, which is defined by λ(n) = (−1) ([1]). • For α 2 N, we have λα = λ t · · · t λ (the unitary convolution of α factors). 130 Thai J. M ath. Special Issue (2020)/ P. Yangklan, V. Laohakosol, P. Ruengsinsub and S. Mavecha 2.2 Properties In this sub-section, we derive a number of properties relating to the functions defined above some of which generalize those in (1.2). Note first that for α; β 2 C n f0g, µα t µβ = µα+β (see [6]). Proposition 2.2.1. For α; β 2 R, n 2 N, p prime, we have α Q ανp(n) 1) σ (n) = pjn p + 1 α 2) ζα = µ t σ Q νp(n) 3) 'α(n) = pjn p − α P Q νp(n) 4) djjn 'α(d) = pjn p + 1 − α β 5) 'α t σ = ζ1 t ζβ t µα−1 ∗ Q ανp(n) 6) 'α(n) = pjn p − 1 P ∗ 7) djjn 'α(d) = ζα(n) ∗ ∗ 8) ζα t 'β = ζβ t 'α ∗ α 9) 'α t d = σ ∗ β 10) 'α t σ = ζα t ζβ P Q νp(n) 11) djjn λα(d) = pjn 1 + (−α) Q νp(n) 12) λα t µβ (n) = pjn (−α) − β β Q βνp(n) νp(n) 13) (λα t σ )(n) = pjn p + (−α) + 1 Q νp(n) νp(n) 14) (λα t 'β)(n) = pjn p + (−α) − β ∗ Q βνp(n) νp(n) 15) (λα t 'β)(n) = pjn p + (−α) − 1 : Proof. 1) Since σα is multiplicative, we get σα(1) = 1. To prove the assertion, it suffices to evaluate α σ (pa) for prime p and a 2 N, which is X σα(pa) = dα = pαa + 1: djjpa Assertions 2), 5), 7), 8), 9) and 10) follow, respectively, from the identities α µ t σ = µ t ζα t µ−1 = ζα β 'α t σ = ζ1 t µα t ζβ t µ−1 = ζ1 t ζβ t µα−1 X ∗ ∗ 'α(d) = ('α t U)(n) = (ζα t µ t µ−1)(n) = ζα(n) djjn ∗ ∗ ζα t 'β = ζα t ζβ t µ = ζβ t ζα t µ = ζβ t 'α ∗ α 'α t d = ζα t µ t µ−2 = ζα t µ−1 = σ ∗ β 'α t σ = ζα t µ t ζβ t µ−1 = ζα t ζβ t I = ζα t ζβ: Unitary Analogues of Some Arithmetic Functions 131 ∗ β ∗ Since the functions 'α;'α; λα t µβ; λα t σ ; λα t µβ; λα t 'β are multiplicative, as in the proof of Assertion 1), Assertions 3), 6), 12), 13), 14) and 15) follow at once from the evaluation a p a a a X ! d a !(1) !(p ) a 'α(p ) = (ζ1 t µα)(p ) = d(−α) = p (−α) + (−α) = p − α djjpa ∗ a a a a aα 'α(p ) = (ζα t µ)(p ) = µ(p ) + ζα(p ) = p − 1 X pa (λ t µ )(pa) = λ (d)µ = λ (pa) + µ (pa) = (−α)a − β α β α β d α β djjpa X pa (λ t σβ)(pa) = λ (d)σβ = λ (pa) + σβ(pa) = (−α)a + pβa + 1 α α d α djjpa X pa (λ t ' )(pa) = λ (d)' = λ (pa) + ' (pa) = (−α)a + pa − β α β α β d α β djjpa X pa (λ t '∗ )(pa) = λ (d)'∗ = λ (pa) + '∗ (pa) = (−α)a + paβ − 1: α β α β d α β djjpa When n = 1, we have X X Ω(1) 'α(d) = 'α(1) = 1; λα(d) = λα(1) = (−α) = 1: djj1 djj1 For n ≥ 2, since the functions 'α t U; λα are multiplicative, as in the proof Assertion 1), Assertions 4) and 11) follow from a X a a a ('α t U)(p ) = 'α(p ) = 'α(1) + 'α(p ) = p + 1 − α djjpa X a Ω(pa) a λα(d) = λα(1) + λα(p ) = 1 + (−α) = 1 + (−α) : djjpa 3 Results Complementing the concept of exponentially odd numbers, an exponentially even number is defined to be a positive integer whose prime factorization contains only even prime powers, and let Ee be the set of all exponentially even numbers.