Bounds for the Density of Abundant Integers

Transcription

1 Bounds fo the Density of Abundant Integes Mac Deléglise CONTENTS Intoduction. Eessing A() as a Sum 2. Tivial Bounds fo A () 3. Lowe Bound fo A() 4. Ue Bounds fo A () 5. Mean Values of f(n) and Ue Bounds fo A () 6. Ue Bounds fo the Eule Poducts 7. Numeical Results 8. Othe Eeimental Results Refeences V () We say that an intege n is abundant if the sum of the divisos of n is at least 2n. It has been nown [Wall 972] that the set of abundant numbes has a natual density A(2) and that.244 < A(2) <.29. We give the shae bounds INTRODUCTION.2474 < A(2) <.248. Let be a ositive eal numbe, and n an intege. Let (n) be the sum of the divisos of n, and set f(n) = (n) ; A () = fn f(n) g ( ) n A numbe in A () is called -abundant, o simly abundant if = 2. Davenot oved that A () has a natual density A(), and that A() is a continuous function of ; see, fo eamle, [Davenot 933; Elliott 979, Chate 5; Tenenbaum 995, III. and III.2]. Behend [933] oved that 24 < A(2) < 34, and Wall [972] imoved this to 244 < A(2) < 29. We ove hee the following Theoem.. The density A(2) of the set of abundant numbes satises 2474 < A(2) < 248 This answes a question ased by Heni Cohen Is the ootion of abundant numbes moe o less than a quate? The method used is essentially that given by Behend, the comute allowing us to do moe comutations. This method in fact gives the density A() fo evey. Pehas it could be wothwile to ty an analytic method. Cohen, Deshouilles, Matinet showed in c A K Petes, Ltd /998 $.5 e age Eeimental Mathematics 72, age 37

2 38 Eeimental Mathematics, Vol. 7 (998), No. 2 [Matinet et al. 973] that the Mellin tansfom of A() is the function g(s) = s 2 s l s l l+ Hence, by invesion, we have fo evey > A() = 2i Z +i i s g(s) ds; but the comutation of this integal seems to be dicult; taing = 2 and = 2 we comuted the sum between 2 i and 2 + i, and got the aoimate value 242. Fo lage values of Im(s) the comutation of g(s) is dicult.. EPRESSING A() AS A SUM We denote by ( n ) n the inceasing sequence of imes. Let be a ed intege. We conside the set A () = fn f(n) ; gcd(n; 2 ) = g ( ) This set has a density [Elliott 979; Tenenbaum 995], which will be denoted by A (). Let n be an abitay intege. We denote by n the oduct of the ime factos of n among f ; 2 ; ; g and we wite n = n n 2. The function f is multilicative and f(n) = f(n )f(n 2 ) is geate than o equal to if and only if f(n 2 ) =f(n ). This oves that A () is atitioned as follows A () = [ n = Consideing the densities we have Poosition.. A() = n = n A f(n ) A ; ( 2) n f(n ) whee the sum is taen ove all n that ae a oduct of imes belonging to f ; 2 ; ; g. To see this, it is sucient to ove the following lemma. Lemma.2. Let be an intege geate than and (A ) a sequence of disjoint sets having densities d. Set A = S A. Then A has a density d(a ) and Poof. Wite A = [ d(a ) = d [ A [ A > The second set in this union is fomed of multiles of +. Its ue density is bounded by = + and d d(a ) d(a ) d + + whee d and d denote the lowe and ue densities. We let! and we get the esult. 2. TRIVIAL BOUNDS FOR A () Poosition 2.. Fo evey and evey > we have and A () F (2 ) A () = F if ; (2 2) whee F = Q i= ( = i). Poof. Clea, since A () is fomed only with integes coime with 2, and comises all these integes if. 3. LOWER BOUND FOR A() Let z be a abitay ositive eal aamete. If in ({2) we just ee the integes n = z, we get a lowe bound fo A(). Hence n z A() n = n A f(n )

3 Deléglise Bounds fo the Density of Abundant Integes 39 We still get a lowe bound if we just ee those n such that f(n ) ; hence A() n z n = f (n ) n A f(n ) n = f (n ) By (2{2), all the A (=f(n )) ae equal to F ; hence n z A() F (3 ) n This lowe bound is almost tivial and could have been shown slightly dieently. We choose an ue bound z and a set f ; 2 ; ; g of small imes. We comute all the integes m less than z, comosed of ime factos fom f ; 2 ; ; g, and -abundant. Evey multile of an abundant numbe being abundant, all the oducts of the numbes m thus obtained by some ime factos out of f ; 2 ; ; g ae still abundant numbes. The lowe bound fo A() is the density of this set, F Pm =m. 4. UPPER BOUNDS FOR A () As in the evious section, we intoduce a eal ositive aamete z and wite A() = n z n = n A f(n ) + z<n n = A n f(n ) In the second sum, each value of A is bounded fom above by F ; thus the second sum is bounded fom above by F z<n n = n = F n n = = F n z n = n F n ; n z n = n so A() n z n = A n f(n ) n z + F n = n (4 ) It emains to bound the values of A that aea in the sum (4{). If we just use the tivial ue bound A F we will get A(), so we need a nontivial ue bound fo A (). This is the subject of the net section. 5. MEAN VALUES OF f(n) AND UPPER BOUNDS FOR A () Let f be the multilicative function that taes the value fo with and the value f( ) fo >. We an intege and we conside g = f and the mean value of g comuted on the st n integes M n = n n m= g(m) Let be the convolution oduct of g and the Mobius function (m) = djm The Mobius invesion fomula gives M n = n = n d= n m= n d= g(n) = n (d) (d) d h n di m d g(d) (5 ) n m= djm n = () n d= (d) (d) n d The function (d)=d is multilicative, so () is also equal to the value of the Eule oduct () = + () + (2 ) + + (5 2) 2

4 4 Eeimental Mathematics, Vol. 7 (998), No. 2 Using the denition equation (5{) of, we have ( ) = g( ) g( ) = when > and >, othewise ( ) =. We etun to the sum nm n = n m= g(m) Let B n be the numbe of integes m between and n such that f (m), o equivalently g(m). We collect the tems of this sum in two classes, st those tems fo which g(m), that ae bounded fom below by, and the othe tems, that ae bounded fom below by. We get B n + n B n nm n n (); dividing by n and letting n! we get B () () ; whee B () is the density of the set of all m such that f (m). This set is the disjoint union of the A (), and we deduce the following ue bound, oved by Behend [933]. Poosition 5.. Fo evey intege and evey, A () F () (5 3) Table gives the ue bounds fo 95 () fo = ; 2; 4; 8; 6; ; () 95 () TABLE. Ue bounds fo 95 (). When is vey close to, almost evey intege is -abundant and the tivial ue bound (2{) is bette than the ue bound (5{3). Table 2 shows this henomenon. It gives fo some values of the best ue bound fo A () obtained by fomula (5{3) choosing the ight value fo. The value = on the st line means that, fo this =, the tivial ue bound (2{) is the bette one. A 95 () A 95 () TABLE 2. Some ue bounds fo A 95 () obtained using Table. 6. UPPER BOUNDS FOR THE EULER PRODUCTS V () In this section we give some eective ue bounds used to get ue bounds fo the Eule oducts (). In all this section we wite ( ) = = dj (d) f d This is the function dened by (5{) fo =. We gave in [Deleglise and Nicolas 994] a method to quicly comute Q a good aoimate value of an Eule oduct g(=), when g is a holomohic function aound whose st Taylo seies coecients ae not too lage. This method could have been used to get some vey accuate values fo the st (). Fo vey lage values of the accuacy would not be so good. Since we just need an ue bound fo each (), we will just use the tivial method nd ue bounds fo the atial oducts, and fo the tails of the oducts. Lemma 6.. Let be an intege and 2. Then + < 3

5 Deléglise Bounds fo the Density of Abundant Integes 4 Poof. We have ( + =) = = e ln( + =) = < e(=) = e=2 =2 < 3 Lemma 6.2. Let an intege 2 and 2. Then < 6 = 9 < 78 Poof. Let u = =. Then y = < = = hence ln(y) = 2u ln u =2u ; u 2 ln = ln 6 9 ; 4 since the function (=u) ln =( u) is inceasing fo < u 2 4. Lemma 6.3. Fo evey intege and evey ime, ( ) + ( + =) Poof. Set + = 4 2 = ; = + We get, fo, ( ) = ( ) = 2 ( ) 2 2 = Using this ue bound fo 2 in the sum we get the conclusion. ( ) Lemma 6.4. Fo evey intege and evey ma(2; 5) we have ( ) < Poof. The eceding thee lemmas give, fo evey 2, ( ) = if 5 Fo = this ue bound is still tue, because ( ) = = Lemma 6.5. Fo evey intege with we have Poof. Set > 6 u = ( ) + 7 > 6 ( ) Using Lemma 6.4 we get ln(u) 3 > 6 2 The sum of = 2 can be comuted as elained in [Deleglise and Nicolas 994,. 33{332], o it can be found in [Glaishe 89] 2 =

6 42 Eeimental Mathematics, Vol. 7 (998), No. 2 Inteval Inteval Inteval Inteval [; ] [; 6 ] [ 9 ; ) [ 4 ; ) [; 2 ] 24 [; 7 ] [ ; + 7 ) [ 5 ; ) [; 3 ] 249 [; 8 ] [ ; + 7 ) [ 6 ; ) [; 4 ] 2492 [; 9 ] [ 2 ; ) [ 7 ; ) [ 3 ; ) [ 8 ; ) TABLE 3. Fequency of abundant numbes in dieent intevals. P Hence, subtacting = 2, we have 6 = > 6 and and nally ln(u) < 9 7 < 3 ; u = e ln u < + using the estimate e t < ; t fo t <. We get an ue bound fo the Eule oduct (5{2), witing > ( ) = < 6 ( ) > 6 ( ) The st oduct is bounded by Lemma 6.3 and the second by Lemma 6.5. Table gives the ue bounds fo 95 () fo = ; 2; 4; 8; 6; ; 496. These ae the values used fo bounding the values A that aea in fomula (4{). in (4{) is bounded using fomula (5{3) with = ; 2; 4; 8; ; 496 and the tivial bound (2{); we ee the best esult obtained. This equies the enumeation of all the not geate than z, which is done by a bactacing ocedue. The total numbe of these n less than 4 whose ime factos ae less than 5 is The comutation was efomed on an HP9-73 wostation, using about hous of CPU time. It yields 2474 < A(2) < 248; (7 ) in aticula A(2) = OTHER EPERIMENTAL RESULTS We comuted the numbe of abundant numbes less than N fo N = ; ; 2 ; ; 9, and also the numbe of abundant numbes in the intevals [N; N + 7 ) fo N = 9 ; ; ; 8. The esults ae given in Table 3 and seem to show that the net digit of A(2) is a 6. We than the efeee fo emaing to us that the numbe of abundant numbes given above in the intevals of size 7 ae comatible with a binomial law with aametes m = and s = NUMERICAL RESULTS We have bounded A(2) using (3{) and (4{) with = 2, = 95 (which is the numbe of imes less than 5), and z = 4. Fo the ue estimate each tem A f( ) REFERENCES [Behend 933] F. Behend, \ Ube Numei abundantes, II", Sitzungsbe. Peuss. Aad. Wiss. (933), 28{ 293. [Davenot 933] H. Davenot, \ Ube Numei abundantes", Sitzungsbe. Peuss. Aad. Wiss. (933), 83{837.

7 Deléglise Bounds fo the Density of Abundant Integes 43 [Deleglise and Nicolas 994] M. Deleglise and J.-L. Nicolas, \Su les enties infeieus a ayant lus de log() diviseus", J. Theo. Nombes Bodeau 62 (994), 327{357. [Elliott 979] P. D. T. A. Elliott, Pobabilistic numbe theoy, I Mean-value theoems, Gundlehen de Mathematischen Wissenschaften, Singe, New o, 979. [Glaishe 89] W. L. Glaishe, \On the sums of the invese owes of the ime numbes", Quately Jounal of Math. 25 (89), 347{362. [Matinet et al. 973] J. Matinet, J. M. Deshouilles, and H. Cohen, \La fonction somme des diviseus",. E. No. in Seminaie de Theoie des Nombes, 972{973 (Univ. Bodeau I, Talence), Lab. Theoie des Nombes, Cente Nat. Recheche Sci., Talence, 973. [Tenenbaum 995] G. Tenenbaum, Intoduction a la theoie analytique et obabiliste des nombes, 2nd ed., Cous secialises, Soc. math. Fance, Pais, 995. [Wall 972] C. R. Wall, \Density bounds fo the sum of divisos function",. 283{287 in The theoy of aithmetic functions (Kalamazoo, Mich., 97), edited by A. A. Gioia and D. L. Goldsmith, Lectue Notes in Math. 25, Singe, Belin, 972. Mac Deleglise, Institut Giad Desagues (UPRES-A 528), Deatement de Mathematiques, Univesite Lyon, 43 Bld du Novembe 98, Villeubanne cede, Fance (deleglis@desagues.univ-lyon.f) Received Januay 2, 997; acceted August, 997