THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION

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1 MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia e-ail: gleboyat@ail.ru Suary. We foud the effect of the deforatio of the uer liit of the divisor fuctio, deedig o the rate of growth of the diesio cocerig the arguet of the divisor fuctio. 010 Matheatics Subject Classificatio: 05A10, 05A16, 11B65, 11B75, 11D57, 11N37, 11N56. Key words ad Phrases: Divisor Fuctio, Icreasig Diesio, Multilicative Fuctio. 17

2 1. INTRODUCTION Let deote ultidietioal divisor fuctio, as usual, the uber of solutios of the equatio x1 x x i ositive iteger x1, x,, x for fix iteger ad. We suose (0) 0, (1) 1, 1. For value of 1 is the uber of distict divisors a ositive iteger. I geeral case of ultidiesioal divisor fuctio the uber we call diesio. I 1907, S. Wigert [1] showed that l x l x l l l x ax x ex l O, l l x (l l x) hece it follows as a cosequece of the existece of a uer liit li l l l su l l. (1) S. Raauja [] i 1915 gave eough iterestig ad a sile roof of this relatio. Siilar techiques ca be geeralized for forula (1) for ay fixed diesio of the divisor fuctio l l l li su l. () l We defie l l l f, where. l l J.L. Nicolas ad G. Robi [3] established that the axiu of f is reached at uber , ad ax f f ( 0 ) Also i the article [3] they roved a shar iequality l C1, l l where 3 ad C a costat that for 0 equality holds. The article [4] gives several iterestig ajorat estiates of the divisor fuctio with 56 : l l C, where C ; l l (l l ) l l l C 3, where C ; l l (l l ) (l l ) l, where C l l C 4 18

3 I the article [5] author obtaied the axial order of the ratio of the uber of [ ] divisors of "adjacet" bioial coefficiets. For iteger 1 ad we roved the shar iequality 1 1 C ( 1) T, (3) C ad for each iteger 1, there is a ifiite uber that i (3) equality holds. I this aer, we ivestigate the behavior of the uer liit of the divisor fuctio o the set of atural ubers with icreasig diesio. If for, the axiu value (i the sese of the uer liit) of the divisor fuctio differ fro classical uer liit () at a sufficietly fast growth of diesio, ad the uer liit is achieved o the ower sequeces. Recall that the behavior of the average value of the divisor fuctio with icreasig diesio is ivestigated. The aer [6] shows that the ai ter of the asytotic forula for the ea value chages accordig to which the values of diesio lie i each of the itervals where is a iteger, 0 3 Let (l x) (l x),, ad x legth of the iterval of averagig. be the -th rie uber ( 1 ). We defie the sequece of ubers. The the uer liit of the ultidiesioal divisor fuctio (i the sese of the uer liit with a fixed diesio) is achieved o a sequece { }. I articular, ( ), fro the law of distributio of rie ubers it follows that l l o o, l l l l l l (4) l l o( ). We use the well-ow exressio for the ultidiesioal divisor fuctio 1 ( ), which by ultilicativity easily exteded to all ositive itegers. Always uder the sybol we ea a rie uber. If we record, we always cosider soe atural uber,. We use the sybols O ad o i their usual sese whe. It should be oted that whe we use iequalities 19

4 F G 1 o(1), we cosider that the araeter taes quite large values, for which iequality F G 1 R holds, where R 0 as. I all such iequalities fuctio R ca be reseted clearly.. PRELIMINARIES AND AUXILIARY RESULTS LEMMA 1. (i) The estiate 1 1 e holds for. (ii) The estiate 1 1 e holds for. PROOF. It suffices to rove oly oe of two oits, as ad aroval of the ites obtaied fro each other by relacig ad.we rove the first iequality. Trasforatio of the bioial coefficiet is 1 ( 1) ( 1) !! Note that for 1 i followig iequality holds Cosequetly, i i e 1,! iasuch as!. Lea is roved. e For a ositive iteger we write ad for the uber of distict rie factors of ad the total uber of rie factors of (icludig ultilicities), resectively. We defie the fuctios ad as follows: 0

5 , 1. LEMMA. Let, ad 0 as. The the followig liit relatio li su 1 holds. PROOF. This follows fro two stateets: - the uer liit is reached o a sequece {( ) }, where 1 a roduct of cosecutive ries, 1 ; - for ay ( ) the iequality holds. Ideed, whe ( ) we see that ad fro (4), we have 1 1 o(1) 1 o(1). Thus, whe ( ) we get li 1. Lea is roved. LEMMA 3. Let, ad 0 relatio li su 1 holds. PROOF. This follows fro two stateets: - the uer liit is reached o a sequece { } cosecutive ries, 1 ; 1 - for ay ( ) the iequality ( ) holds. The roof is siilarly to Lea. 3. PROOFS OF MAIN RESULTS THEOREM 1. Let ad holds as. The the followig liit, where 1 a roduct of 0 as. The the followig equality 1

6 su 1. li PROOF. We show that the uer liit is achieved o a sequece { } where 1 the roduct of cosecutive ries, 1. We have ( ). 1 1 The exressio (4) gives us for the liitig relatio li 1. Further it is ecessary to show that for the iequality is satisfied: 1 o(1). We write the divisor fuctio i the for of two ieces: (5) We estiate the arith of the first roduct by usig Lea 1: 1 e l l Accordig to Lea we fid that 1 1 o(1). (6) Oce agai use of Lea 1 gives us the estiatio of the secod iece: 1 1 e. l

7 Let us deote ˆ ad N, where 1 ˆ. The we get 1 ( N), l N l ad ˆ N ˆ iasuch as 1. have Lea 3 ilies that N ( N) 1 o(1). N o. So, uttig together estiates (6) ad (7), we obtai the desired iequality (4). Fro the relatio (1) we The theore is roved. THEOREM. Let ad 0 whe. The the followig equality holds li su 1. PROOF. First of all, we ote that uder the coditios of the theore for ay rie divisor iequality holds. Therefore, to estiate the divisor fuctio, we ca sily use oly the secod iequality i Lea 1: 1 1. l It is sufficiet to rove that 1 o(1). Ideed, usig the obvious iequality, we have 1 ( 1) ( 1o(1) ). It reais to show that the uer liit secified i the theore is obtaied. For this urose we cosider the sequece. The we get ˆ (7) 3

8 1 1 j! j1 1 e.! l Sice 0 as the ( ) li li l 1. Taig ito accout that, we get The theore is roved. 4. ACKNOWLEDGEMENTS ( ) li 1. The author wishes to exress dee gratitude to Professor V.N. Chubariov for his careful attetio of this wor. This research was artly suorted by RFFI (grat o ). REFERENCES [1]. S. Wigert, Sur l'ordre de gradeur du obre des diviseurs d'u etier, Ariv for Mateati, Astrooi och Fysi 3 (1907), 1-9. []. S. Raaua, Highly coosite ubers, Proc. Lodo Math. Soc. (), Vol. 14 (1915), [3]. J.L. Nicolas ad G. Robi, Majoratios exlicites our le obre de diviseurs de, Caad. Math. Bull. 6 (1983), [4]. G. Robi, These d'etat, Uiversite de Lioges, Frace, [5]. G.V. Fedorov, O the uber of divisors of bioial coefficiets, Matheatical Notes, 93 (013), (Origial Russia Text: G. V. Fedorov, 013, ublished i Mateatichesie Zaeti, 013, Vol. 93,, ). [6]. G.V. Fedorov, O a theore of A.I. Pavlov, Dolady Matheatics, 86 (01), (Origial Russia Text: G.V. Fedorov, 01, ublished i Dolady Aadeii Nau, 01, Vol. 446, 3, ). Received October 1, 013 4