EXTREMAL ORDERS OF COMPOSITIONS OF CERTAIN ARITHMETICAL FUNCTIONS. Abstract

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1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 EXTREMAL ORDERS OF COMPOSITIONS OF CERTAIN ARITHMETICAL FUNCTIONS József Sádor Babeş-Bolyai Uiversity, Departmet of Mathematics ad Computer Scieces, Str. Kogăliceau Nr. 1, Cluj-Napoca, Romaia jjsador@hotmail.com László Tóth Uiversity of Pécs, Istitute of Mathematics ad Iformatics, Ifjúság u. 6, 7624 Pécs, Hugary ltoth@ttk.pte.hu Received: 2/7/08, Revised: 7/15/08, Accepted: 7/27/08, Published: 7/30/08 Abstract We study the exact extremal orders of compositios f(g()) of certai arithmetical fuctios, icludig the fuctios σ(), φ(), σ () ad φ (), represetig the sum of divisors of, Euler s fuctio ad their uitary aalogues, respectively. Our results complete, geeralize ad refie kow results. 1. Itroductio Let σ(), φ() ad ψ() deote as usual the sum of divisors of, Euler s fuctio ad the Dedekid fuctio, respectively, where ψ() = p (1 + 1/p). Extremal orders of the composite fuctios σ(φ()), φ(σ()), σ(σ()), φ(φ()), φ(ψ()), ψ(φ()), ψ(ψ()) were ivestigated by L. Alaoglu ad P. Erdős [1], A. M akowski ad A. Schizel [9], J. Sádor [10], F. Luca ad C. Pomerace [7], J.-M. de Koick ad F. Luca [8], ad others. For example, i paper [9] it is show that (1) lim if σ(σ()) = 1, (2) lim sup φ(φ()) = 1 2,

2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 2 while i paper [7] the result (3) lim sup is proved, where γ is Euler s costat. σ(φ()) log log = eγ It is the aim of the preset paper to exted the study of exact extremal orders to other compositios f(g()) of arithmetical fuctios, cosiderig also the fuctios σ () ad φ (), represetig the sum of uitary divisors of ad the uitary Euler fuctio, respectively. Recall that d is a uitary divisor of if d ad (d, /d) = 1. The fuctios σ () ad φ () are multiplicative ad if = p a 1 1 p ar r is the prime factorizatio of > 1, the (4) σ () = (p a ) (p ar r + 1), φ () = (p a 1 1 1) (p ar r 1). Note that σ () = σ(), φ () = φ() for all squarefree ad that for every 1, (5) φ() φ () σ () ψ() σ(). We give some geeral results which ca be applied easily also for other special fuctios. Our results complete, geeralize ad refie kow results. They are stated i Sectio 2, their proofs are give i Sectio 3. Some ope problems are formulated i Sectio Mai Results Theorem 1. Let f be a arithmetical fuctio. Assume that (i) f is itegral valued ad f() 1 for every 1, (ii) f() for every sufficietly large ( 0 ), (iii) f(p) = p 1 for every sufficietly large prime p (p p 0 ). The (6) lim sup σ(f()) log log σ(f()) f() log log f() = eγ, (7) lim sup ψ(f()) log log ψ(f()) f() log log f() = 6 π 2 eγ, (8) lim sup σ(f()) φ(f())(log log ) 2 σ(f()) φ(f())(log log f()) 2 = e2γ,

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 3 (9) lim sup ψ(f()) φ(f())(log log ) 2 ψ(f()) φ(f())(log log f()) 2 = 6 π 2 e2γ. Theorem 1 ca be applied for f() = φ() ad f() = φ (), the uitary Euler fuctio. For example, (6) ad (7) give (10) lim sup σ(φ ()) log log = eγ, (11) lim sup ψ(φ()) log log = 6 π 2 eγ. The weaker result lim sup ψ(φ()) = is proved i [10]. Figure 1 is a plot of the fuctios σ(φ ()) ad e γ log log for Theorem 2. Let g be a arithmetical fuctio. Assume that (i) g is itegral valued ad g() 1 for every 1, (ii) g() for every sufficietly large ( 0 ), (iii) either g(p) = p + 1 for every sufficietly large prime p (p p 0 ), or g is multiplicative ad g(p) = p for every sufficietly large prime p (p p 0 ). The (12) lim if φ(g()) log log = lim if φ(g()) log log g() g() = e γ. Theorem 2 applies for g() = σ(), σ (), ψ(), σ (e) (), where σ (e) () represets the sum of expoetial divisors of. We have for example (13) lim if φ(σ()) log log = e γ. φ(σ()) Remark that accordig to a result of L. Alaoglu ad P. Erdős [1], lim set of desity 1. = 0 o a Theorems 1 ad 2 ca be geeralized as follows. If f() 1 is a iteger valued arithmetical fuctio, let f k () deote its k-fold iterate, i.e., f 0 () =, f 1 () = f(),..., f k () = f(f k 1 ()).

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 4 Figure 1: Plot of σ(φ ()) ad e γ log log for Theorem 3. Let f be a arithmetical fuctio. Suppose that (i) f is itegral valued ad 1 f() for every 1, (ii) f(p) = p 1 for every prime p, (iii) for every s, t 1 if s t, the f(s) f(t). The for every k 0, (14) lim sup σ(f k ()) f k () log log = eγ. Theorem 3 applies for f() = φ(), f() = (p 1 1) (p r 1), f() = (p 1 1) a1 (p r 1) ar, where = p a 1 1 p ar r. Theorem 4. Let g be a arithmetical fuctio. Suppose that (i) g is itegral valued ad g() for every 1, (ii) g(p) = p + 1 for every prime p,

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 5 (iii) for every s, t 1 if s t, the g(s) g(t). The for every k 0, (15) lim if φ(g k ()) log log g k () = e γ. Theorem 4 applies for g() = ψ(), g() = (p 1 + 1) (p r + 1), g() = (p 1 + 1) a1 (p r + 1) ar, where = p a 1 1 p ar r. For f() = φ() ad g() = ψ() we have for every k 0, (16) lim sup (17) lim if σ(φ k ()) φ k () log log = eγ, φ(ψ k ()) ψ k () log log = e γ. Compare Theorems 1 4 with the followig deep results: σ k () for k 2 the ormal order of σ k 1 () is keγ log log log, i.e. σ k () ke γ σ k 1 () log log log o a set of desity 1, cf. P. Erdős [2], φ k () for k 1 the ormal order of φ k+1 () is keγ log log log, proved by P. Erdős, A. Graville, C. Pomerace ad C. Spiro [4]. φ(σ()) the ormal order of is e γ / log log log ad the ormal order of σ() e γ log log log, see L. Alaoglu ad P. Erdős [1]. σ(φ()) φ() is Note that the average orders of φ()/φ 2 () ad φ 2 ()/φ() were ivestigated by R. Warlimot [15]. Theorem 5. Let h() be a arithmetical fuctio such that h() σ() for every sufficietly large ( 0 ). The (18) lim if h(σ()) = 1. For h() = σ() this is formula (1), for h() = ψ() it is due by J. Sádor [10], Theorem Theorem 5 applies also for h() = σ (), σ (e) (). Theorem 6. (19) lim sup φ(φ ()) φ (φ()) φ (φ ()) = 1.

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 6 Compare the results of (19) with (2). Figure 2 is a plot of the fuctios φ (φ()) ad for Figure 2: Plot of φ (φ()) ad for Cocerig φ (φ ()) ad σ (φ ()) we also prove: Theorem 7. (20) lim if φ (φ ()) log log log > 0. Theorem 8. (21) lim if σ (φ ()) { σ (φ (m/2)) if m/2 } : 2 m, m 2 l, l 2, (22) lim if where ε = 3 4(2 32 1) σ (φ ()) ε,

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A Proofs The proofs of Theorems 1 ad 2 are similar to the proof of (3) give i [7], usig a simple argumet based o Liik s theorem, which states that if (k, l) = 1, the there exists a prime p such that p l (mod k) ad p k c, where c is a costat (oe ca take c 11). Proof of Theorem 1. To obtai the maximal orders of the fuctios σ()/, ψ()/, σ()/φ() ad ψ()/φ(), which are eeded i the proof, we apply the followig result of L. Tóth ad E. Wirsig, see [13], Corollary 1: If F is a oegative real-valued multiplicative arithmetic fuctio such that for each prime p, a) ρ(p) := sup ν 0 F (p ν ) (1 1/p) 1, ad b) there is a expoet e p = p o(1) satisfyig F (p ep ) 1 + 1/p, the lim sup F () log log = eγ p ( 1 1 ) ρ(p). p For F () = σ()/ (with ρ(p) = (1 1/p) 1, e p = 1), F () = ψ()/ (with ρ(p) = 1 + 1/p, e p = 1), F () = σ()/φ() (with ρ(p) = (1 1/p) 1, e p = 1) ad F () = ψ()/φ() (with ρ(p) = (p + 1)/(p 1), e p = 1), respectively, we obtai (23) lim sup (24) lim sup (25) lim sup σ() log log = eγ, ψ() log log = 6 π 2 eγ, σ() φ()(log log ) 2 = e2γ, (26) lim sup ψ() φ()(log log ) 2 = 6 π 2 e2γ. Note that (23) is the result of T. H. Growall [5], (26) is due to S. Wigert [16] ad (25) is better tha lim sup σ()/φ() = give i [11]. We ow prove (6). Usig assumptio (ii), l f : σ(f()) log log l f : σ(f()) lim sup f() log log f() m σ(m) m log log m = eγ,

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 8 accordig to (23). For every, let p be the least prime such that p p 0 ad p 1 (mod ). Here p 1 ad by Liik s theorem p c, so log log p log log. Hece, usig coditio (iii), σ(f(p )) = σ(p 1) p log log p p log log p σ(p 1) (p 1) log log σ() log log, applyig that if s t, the σ(s)/s = d s 1/d d t 1/d = σ(t)/t. We obtai that l f e γ, therefore e γ l f l f eγ, that is l f = l f = eγ. The proofs of (7), (8), (9). Aalogous to the method of above takig ito accout (24), (25), (26) ad that s t implies ψ(s)/s ψ(t)/t, σ(s)/φ(s) σ(t)/φ(t), ψ(s)/φ(s) ψ(t)/φ(t). Proof of Theorem 2. This is similar to the proof of Theorem 1. We use a result of E. Ladau [6], (27) lim if φ() log log = e γ. By coditio (ii) ad usig that the fuctio (log log x)/x is decreasig for x x 0, l g := lim if accordig to (27). φ(g()) log log l g := lim if φ(g()) log log g() g() lim if m φ(m) log log m m = e γ, Assume that g(p) = p+1 for every p p 0. For every, let q be the least prime such that q p 0 ad q 1 (mod ). Here q + 1 ad by Liik s theorem log log q log log. Hece φ(g(q )) log log q q = φ(q + 1) log log q q φ(q + 1) log log q + 1 φ() log log, applyig that if s t, the φ(s)/s φ(t)/t. We obtai that e γ l g, therefore e γ l g l g e γ, that is l g = l g = e γ. Now suppose that g is multiplicative ad g(p) = p for every prime p p 0. As it is kow, i (27) the limif is attaied for = k = p 1 p k, the product of the first k primes, whe k. Sice g( k ) = g(p 1 p k ) = g(p 1 ) g(p k ) = p 1 p k = k, φ(g( lim k )) log log k φ( k k = lim k ) log log k k k = e γ. Proof of Theorem 3. By coditio (i), f 2 () = f(f()) f() ad f k () for every k 0. Therefore, l k : σ(f k ()) lim sup f k () log log σ(f k ()) f k () log log f k () l 0 : m σ(m) m log log m = eγ,

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 9 by (23), for every k 0. By (iii), if s t, the f(s) f(t), f 2 (s) f 2 (t) ad f k (s) f k (t) for every k 0. Now let k 1. If p is the least prime such that p 1 (mod ), cf. proof of Theorem 1, the p 1 ad f k 1 () f k 1 (p 1). Therefore, applyig also (ii), σ(f k (p )) σ(f k 1(p 1)) f k (p ) log log p f k 1 (p 1) log log σ(f k 1()) f k 1 () log log = l k 1, Hece l k l k 1, ad it follows l k l k 1... l 0, l 0 l k l 0, l k = l 0 = e γ. Proof of Theorem 4. Similar to the proof of Theorem 3. By coditio (i), g 2 () = g(g()) g() ad g k () for every k 0. Therefore, L k := lim if φ(g k ()) log log g k () lim if φ(g k ()) log log g k () g k () L 0 : m φ(m) log log m m = e γ, by (27), for every k 0. By (iii), if s t, the g(s) g(t), g k (s) g k (t) for every k 0. Now let k 1. If q is the least prime such that q 1 (mod ), cf. proof of Theorem 2, the q + 1 ad g k 1 () g k 1 (q + 1). Therefore, applyig also (ii), φ(g k (q )) log log q g k (q ) φ(g k 1(q + 1)) log log g k 1 (q + 1) φ(g k 1()) log log g k 1 () = L k 1, Hece L k L k 1, ad it follows L k L k 1... L 0, L 0 L k L 0, L k = L 0 = e γ. Proof of Theorem 5. By h() we have h(σ()) σ(), h(σ())/ 1 ( 0 ). We use that for a fixed iteger a > 1 ad with p prime, for N(a, p) = ap 1 ad for a a 1 arithmetical fuctio satisfyig φ() F () σ() ( 0 ) oe has F (N(a, p)) (28) lim = 1, p N(a, p) cf., for example, D. Suryaarayaa [12]. For p, q primes, σ(q p 1 ) = qp 1 q 1 = N(q, p). We obtai, usig (28), h(σ(q p 1 )) q p 1 = h(n(q, p)) N(q, p)) q p 1 q p 1 (q 1) q, as p, q 1 where q q 1 < 1 + ɛ for each ɛ > 0 if q q(ɛ). Proof of Theorem 6. We have φ() ad φ () for all 1, ad hece φ(φ ()) φ (). Similarly, φ (φ ()).

10 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 10 If = 2 p, p prime, the φ () = 2 p 1 ad φ(φ ()) usig (28) for a = 2 ad F () = φ(). = φ(2p 1) = φ(2p 1) 2p 1 1, p, 2 p 2 p 1 2 p Similarly the relatio holds for φ (φ ()), usig (28) for F () = φ (). For φ (φ()) this ca ot be applied ad we eed a special treatmet. Let M = { [2 log x], if p < x p ap, where a p = 1/2, 4, if p [x 1/2 (p prime)., x] p x Let q be the least prime of the form q M + 1 (mod M 2 ). By Liik s theorem oe has q M c, where c satisfies c 11. Now, put = q. The φ() = q 1 = M(1 + km) = MN for some k. Thus (M, N) = 1, so N is free of prime factors x. Sice φ is multiplicative, φ (φ()) = φ (M) M φ (N) N MN 1 + MN. Here MN 1 + MN 1, as = q, so it is sufficiet to study φ (M) M ad φ (N) N. Clearly, φ (M) M = p ap 1 = ( 1 1 ). If p < x 1/2, the p ap 2 [2 log x] > x for p ap p ap p x p x sufficietly large x. Otherwise, p ap (x 1/2 ) 4 = x 2 > x agai. So p ap > x ayway, implyig that (29) ( ϕ (M) M > 1 x) 1 π(x) ( ) 1 = 1 + O. log x Remark that M < p 2 log x p 4 < exp ( O(x 1/2 log x + x) ) = exp ( O(x) ) by the p<x 1/2 p x well-kow fact: p = e O(a). From q M c ad M < exp ( O(x) ), by N M 10 it follows p a also that (30) N < exp ( O(x) ). (log x) Thus (31) Let ow N = k i=1 q b i i be the prime factorizatio of N. We have log N = k b i, as q i > x for all 1 i k. Here i=1 φ (N) N = k i=1 k i=1 b i k, thus k < log N log x k b i log q i > i=1 x log x ( 1 1 ) ( > 1 1 k ( 1 q b i x) 1 O(x/ log x) ( ) 1 = 1 + O. x) log x i by (30).

11 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 11 By (29) ad (31), φ (φ()) we get log x, so φ (φ()) As φ (φ()) φ() ( ) 1 > 1 + O for sufficietly large. As exp ( O(x) ), log x 1, as = q. 1, the proof is ready. Proof of Theorem 7. For all 1, φ () P () 1, where P () is the greatest prime factor of. Let = 2 p, p prime, the φ (φ ()) = φ (2 p 1) P (2 p 1) 1. Now we use the followig result of P. Erdős ad T. N. Shorey [3]: P (2 p 1) cp log p for every prime p, where c > 0 is a absolute costat, ad obtai (32) ad the result follows. φ (φ ()) log log log cp log p 1 p log 2(log p + log log 2) c log 2, p, Proof of Theorem 8. To prove (21), remark that if 2 m ad m 2 l (l 2), the m/2 is ot a power of 2, so φ (m/2) will be eve (havig at least a odd prime divisor). Sice 2 φ (m/2), oe ca write σ (2φ (m/2)) < 2σ (φ (m/2)). Let p be a sufficietly large prime (p > p 0 ), the (p, m/2) = 1 ad obtai by the above remark. σ (φ (mp/2)) mp/2 σ ((p 1)/2)σ (2φ (m/2)) mp/2 = σ ((p 1)φ (m/2)) mp/2 σ ((p 1)/2) p/2 σ (φ (m/2)) m/2 F ((p 1)/2) It is kow that 1, as p, for F () = σ(), see [9] ad it follows (p 1)/2 that it holds also for F () = σ () ad obtai (21). Now for (22) let m = 4(2 32 1) = 4F 0 F 1 F 2 F 3 F 4 be 4 times the product of the kow Fermat primes. The φ (m/2) = φ (2F 0 F 1 F 2 F 3 F 4 ) = = 2 31, σ (φ (m/2)) m/2 = (2 32 1) = 1 + ε, with the give value of ε Ope Problems Problem 1. Are the results of Theorem 1 valid if f() for each 0 ad f(p) = p for each prime p p 0? Let = p ν 1 1 p νr r > 1 be a iteger. A iteger a is called regular (mod ) if there is a iteger x such that a 2 x a (mod ). Let ϱ() deote the umber of regular itegers a (mod

12 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 12 ) such that 1 a. Here ϱ() = (φ(p ν 1 1 ) + 1) (φ(p νr r ) + 1), i particular ϱ(p) = p for every prime p, cf. L. Tóth [14]. Does Theorem 1 hold for f() = ϱ()? Problem 2. The method of proof of Theorems 1 4 does ot work i the cases of σ (φ()) ad σ (φ ()), for example. We have lim sup σ (φ()) lim sup log log σ (φ()) lim sup φ() log log φ() σ () log log = 6 π 2 eγ, cf. [13], but the secod part of the proof ca ot be applied, because m does ot imply σ ()/ σ (m)/m. What are the maximal orders σ (φ()) ad σ (φ ())? Figure 3 is a plot of the fuctio σ (φ()) for Figure 3: Plot of σ (φ()) for Problem 3. Note that lim sup σ (σ()) σ(σ ()) σ (σ ()) =,

13 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 13 sice for = k = p 1 p k (the product of the first k primes), σ (σ( k )) k σ( k) k = (1 + 1/p 1 ) (1 + 1/p k ), k ; similarly, the other relatios hold. What are the maximal orders of σ(σ ()), σ (σ()), σ (σ ())? Problem 4. Also, lim if φ(φ ()) = lim if φ (φ()) = lim if φ (φ ()) which follow at oce by takig = k = p 1 p k. Here φ (φ( k )) = φ ((p 1 1) (p k 1)) (p 1 1) (p k 1) 1, ad hece φ (φ( k )) (p ) ) 1 1) (p k 1) 1 < (1 1p1 (1 1pk 0, k, k p 1 p k ad similarly for the other relatios. What are the miimal orders of φ(φ ()), φ (φ()), φ (φ ())? = 0, 5. Maple Notes The plots were produced usig Maple. The fuctios σ () ad φ () were geerated by the followig procedures: sigmastar:= proc() local x, i: x:= 1: for i from 1 to ops(ifactors()[ 2 ]) do p_i:=ifactors()[2][i][1]: a_i:=ifactors()[2][i][2]; x := x*(1+p_i^(a_i)): od: RETURN(x) ed; # sum of uitary divisors phistar:= proc() local x, i: x:= 1: for i from 1 to ops(ifactors()[ 2 ]) do p_i:=ifactors()[2][i][1]: a_i:=ifactors()[2][i][2]; x := x*(p_i^(a_i)-1): od: RETURN(x) ed; # uitary Euler fuctio Ackowledgemets. The authors wish to thak the referee for suggestios o improvig earlier versios of Theorems 1 ad 2, as well as for suggestig a correctio for a iitial versio of Theorems 3 ad 4. The authors thak also Professor Floria Luca for helpful correspodece.

14 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A34 14 Refereces [1] L. Alaoglu, P. Erdős, A cojecture i elemetary umber theory, Bull. Amer. Math. Soc., 50 (1944), [2] P. Erdős, Some remarks o the iterates of the ϕ ad σ fuctios, Colloq. Math., 17 (1967), [3] P. Erdős, T. N. Shorey, O the greatest prime factor of 2 p 1 for a prime p ad other expressios, Acta Arith., 30 (1976), [4] P. Erdős, A. Graville, C. Pomerace, C. Spiro, O the ormal behavior of the iterates of some arithmetic fuctios, i Aalytic umber theory, Proc. Coferece i hoor of Paul T. Batema, Birkhäuser, Bosto, 1990, [5] T. H. Growall, Some asymptotic expressios i the theory of umbers, Tras. Amer. Math. Soc., 14 (1913), [6] E. Ladau, Hadbuch der Lehre vo der Verteilug der Primzahle, Teuber, Leipzig Berli, [7] F. Luca, C. Pomerace, O some problems of M akowski-schizel ad Erdős cocerig the arithmetical fuctios φ ad σ, Colloq. Math., 92 (2002), [8] J.-M. de Koick, F. Luca, O the compositio of the Euler fuctio ad the sum of divisors fuctio, Colloq. Math., 108 (2007), [9] A. M akowski, A. Schizel, O the fuctios ϕ() ad σ(), Colloq. Math., 13 ( ), [10] J. Sádor, O the compositio of some arithmetic fuctios, II., J. Iequal. Pure Appl. Math., 6 (2005), Article 73, 17 pages. [11] B. S. K. R. Somayajulu, The sequece σ()/φ(), Math. Studet, 45 (1977), [12] D. Suryaarayaa, O a class of sequeces of itegers, Amer. Math. Mothly, 84 (1977), [13] L. Tóth, E. Wirsig, The maximal order of a class of multiplicative arithmetical fuctios, Aales Uiv. Sci. Budapest., Sect. Comp., 22 (2003), , see [14] L. Tóth, Regular itegers (mod ), Aales Uiv. Sci. Budapest., Sect. Comp., 29 (2008), , see [15] R. Warlimot, O the iterates of Euler s fuctio, Arch. Math., 76 (2001), [16] S. Wigert, Note sur deux foctios arithmètiques, Prace Mat.-Fiz., 38 (1931),