EXPONENTIAL AND INFINITARY DIVISORS ANDREW V. LELECHENKO Abstract. Our paper is devoted to several problems from the field of modified divisors: namely exponential and infinitary divisors. We study the behaviour of modified divisors, sum-of-divisors and totient functions. Main results con- cern with the asymptotic behaviour of mean values and explicit estimates of extremal orders. 1. Introduction Let m be an exponential divisor (or e-divisor) of n (denote m |(e) n) if m | n and for each prime p | n we have a | b, where pa || m, pb || n. This concept, introduced (e) by Subbarao [18], leads us to the e-divisor function τ (n)= m|(e)n 1 (sequence (e) A049419 in OEIS [20]) and sum-of-e-divisors function σ (n) = (e) m (se- P m| n quence A051377). These functions were studied by many authors, including among P others Wu and P´etermann [17, 25]. Consider a set of arithmetic functions A, a set of multiplicative prime-indepen- dent functions MP I and an operator E : A→MP I such that (Ef)(pa)= f(a). One can check that τ (e) = Eτ, but σ(e) 6= Eσ. Section 3 is devoted to the latter new function Eσ. On contrary several authors, including T´oth [22, 24] and P´etermann [16], studied exponential analogue of the totient function, defining φ(e) = Ef. However φ(e) lacks many significant properties of φ: it is prime-independent and φ(e) ≪ nε. In Section 4 we construct more natural modification of the totient function, which will be denoted by f(e). One can define unitary divisors as follows: m |∗ n if m | n and gcd(m,n/m)=1. Further, define bi-unitary divisors: m |∗∗ n if m | n and greatest common unitary divisor of m and n/m is 1; define tri-unitary divisors: m |∗∗∗ n if m | n and greatest common bi-unitary divisor of m and n/m is 1; and so on. It appears that this process converges to the set of so-called infinitary divisors (or ∞-divisors): m |∞ n arXiv:1405.7597v2 [math.NT] 21 Jun 2014 if m | n and for each p | n, pa || m, pb || n, the binary digits of a have zeros in all places, where b’s have. This notation immediately induces ∞-divisor function τ ∞ (sequence A037445) and sum-of-∞-divisors function σ∞ (sequence A049417). See Cohen [1]. Recently Minculete and T´oth [15] defined and studied an exponential analogue of unitary divisors. We introduce e-∞-divisors: m |(e)∞ n if m | n and for each p | n, a b ∞ ∞ p || m, p || n, we have a | b. In Section 5 we improve an estimate for n6x τ (n) by Cohen and Hagis [2] and briefly examine τ (e)∞. Section 6 is devoted to e-∞- P perfect numbers such that σ(e)∞(n)=2n. 2010 Mathematics Subject Classification. 11A25, 11N37, 11N56. Key words and phrases. Exponential divisors, infinitary divisors, modified sum-of-divisors function, modified totient function. 1 2 ANDREW V. LELECHENKO 2. Notations Letter p with or without indexes denotes a prime number. Notation pa || n means that pa | n, but pa+1 ∤ n. We write f⋆g for Dirichlet convolution (f⋆g)(n)= f(d)g(n/d). Xd|n In asymptotic relations we use ∼, ≍, Landau symbols O and o, Vinogradov symbols ≪ and ≫ in their usual meanings. All asymptotic relations are given as an argument (usually x) tends to the infinity. Letter γ denotes EulerMascheroni constant. Everywhere ε> 0 is an arbitrarily small number (not always the same even in one equation). As usual ζ(s) is the Riemann zeta-function. Real and imaginary components of the complex s are denoted as σ := ℜs and t := ℑs, so s = σ + it. We abbreviate llog x := log log x, lllog x := log log log x, where log x is a natural logarithm. Let τ be a divisor function, τ(n)= d|n 1. Denote τ(a1,...,ak;Pn)= 1 a da1 ···d k =n 1 Xk and τk = τ(1,..., 1; ·). Then τ ≡ τ2 ≡ τ(1, 1; ·). k times Now let ∆(a1,...,ak; x) be an error term in the asymptotic estimate of the | {z } sum n6x τ(a1,...,ak; n). (See [11] for the form of the main term.) For the sake of brevity denote ∆ (x) = ∆(1,..., 1; x). P k k times Finally, θ(a1,...,ak) denotes throughout our paper a real value such that | {z } θ(a1,...,ak)+ε ∆(a1,...,ak; x) ≪ x and we write θk for the exponent of x in ∆k(x). 3. Values of Eσ Theorem 1. log Eσ(n) llog n log 3 (1) lim sup = . n→∞ log n 2 Proof. Theorem of Suryanarayana and Sita Rama Chandra Rao [19] shows that log Eσ(n) llog n log σ(n) lim sup = sup . n→∞ log n n>1 n The supremum can be split into two parts: we have log σ(n) log σ(2) log 3 max = = n66 n 2 2 and for n> 6 we apply estimate of Ivi´c[7] (2) σ(n) < 2.59n llog n to obtain log σ(n) log 2.59 + log n + lllog n < := f(n), n n where f is a decreasing function for n> 6 and f(7) < (log 3)/2. Thus log σ(n) log 3 sup = . n>1 n 2 EXPONENTIALANDINFINITARYDIVISORS 3 Equation (1) shows that Eσ(n) ≪ nε. Theorem 2. 1/2 1/3 (3) Eσ(n)= C1x + (C2 log x + C3)x + C4x + E(x), 6 nXx where C1, C2, C3, C4 are computable constants and x1/5 ≪ E(x) ≪ x1153/3613+ε. ∞ −s Proof. Let F (s)= n=1 Eσ(n)n . We have utilizing (2) ∞ P ∞ F (s)= Eσ(pa)p−as = 1+ σ(a)p−as = p a=0 p a=1 Y X Y X = 1+ p−s +3p−2s +4p−3s +7p−4s + O(pε−5s) = p Y 1+ O(pε−5s) = , (1 − p−s)(1 − p−2s)2(1 − p−3s) p Y so (4) F (s)= ζ(s)ζ2(2s)ζ(3s)H(s), where series H(s) converges absolutely for σ> 1/5. Equation (4) shows that Eσ = τ(1, 2, 2, 3; ·) ⋆ h, 1/5+ε where n6x |h(n)| ≪ x . We apply the result of Kr¨atzel [12, Th. 3] together with Huxley’s [6] exponent pair k = 32/205 + ε, l = k +1/2 to obtain P 1/2 1/3 1153/3613+ε τ(1, 2, 2, 3; n)= B1x + (B2 log x + B3)x + B4x + O(x ) 6 nXx for some computable constants B1, B2, B3, B4. Now convolution argument certi- fies (3) and the upper bound of E(x). The lower bound for E(x) follows from the theorem of K¨uhleitner and Nowak [14]. Theorem 3. 2 1/2 (Eσ(n)) = Dx + P7(log x)x + E(x), 6 nXx where D is a computable constant, P7 is a polynomial with deg P =7 and x4/17 ≪ E(x) ≪ x8/19+ε. Proof. We have ∞ ∞ (Eσ(n))2n−s = 1+ (σ(a))2p−as = n=1 p a=1 X Y X = 1+ p−s +9p−2s + O(pε−3s) = p Y 1+ O(pε−3s) = = ζ(s)ζ8(2s)G(s), (1 − p−s)(1 − p−2s)8 p Y where series G(s) converges absolutely for σ> 1/3. 4 ANDREW V. LELECHENKO Ω-estimate of the error term E(x) follows again from [14]. To obtain E(x) ≪ ≪ x8/19 we use [11, Th. 6.8], which implies 1 8 θ(1, 2, 2, 2, 2, 2, 2, 2, 2) 6 6 . 1+2 − θ8 19 Here we used the estimate of Heath-Brown θ8 6 5/8 [21, p. 325]. 4. Values of f(e) For the usual M¨obius function µ, identity function id and unit function 1 we have τ = 1 ⋆ 1, id = 1 ⋆ µ, σ = 1 ⋆ id . Subbarao introduced in [18] the exponential convolution ⊙ such that for multi- plicative f and g their convolution f ⊙ g is also multiplicative with (5) (f ⊙ g)(pa)= f(pd)g(pa/d). Xd|a For function µ(e) = Eµ and defined in Section 1 functions τ (e) and σ(e) we have τ (e) = 1 ⊙ 1, id = 1 ⊙ µ(e), σ(e) = 1 ⊙ id . This leads us to the natural definition of f(e) = µ(e) ⊙ id (similar to usual φ = = µ⋆ id). Then by definition (5) f(e)(pa)= µ(a/d)pd. Xd|a Let us list a few first values of f(e) on prime powers: a 12 3 4 5 f(e)(pa) p p2 − p p3 − p p4 − p2 p5 − p Note that f(e)(n)/n depends only on the square-full part of n. Trivially f(e)(n) lim sup =1 n→∞ n and one can show utilizing Mertens formula (cf. [5, Th. 328]) that f(e)(n) llog n lim inf = e−γ. n→∞ n Instead we prove an explicit result. Theorem 4. For any n> 44100 f(e)(n) llog n (6) > Ce−γ , C =0.993957. n EXPONENTIALANDINFINITARYDIVISORS 5 Proof. Denote for brevity f(n)= f(e)(n) llog n/n. Let s(n) count primes, squares of which divide n: s(n)= p2|n 1. Let pk denote the k-th prime: p =2, p =3 and so on. One can check that 1 2 P f(n) 6 f p2 . 6 p Yps(n) Now we are going to prove that for every x > 11 inequality (6) holds for n = 2 = p6x p . On such numbers we have Q f(n)= (1 − p−1) · log 2 log p 6 6 pYx pXx and our goal is to estimate the right hand side from the bottom .