SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY

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1 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY K. Soh Reddy, V. Kvih d V. Lkhmi Nry Vrdhm College of Egieerig, Shmhbd, Hyderbd, Adhr Prdeh, Idi E-Mil: ureddyk@gmil.com ABSTRACT Thi ricle commece wih defiiio of low icreig fucio d move o o deliee few properie of low icreig fucio. Beide, everl pplicio i ome problem of umber heory uig he heory of low icreig fucio re lo preeed o how how ueful hee fucio prove i olvig comple problem. Keyword: low icreig fucio, ympoiclly equivle, equece of poiive ieger.. INTRODUCTION Slow icreig fucio re defied follow:.. Defiiio Le f :, ) ( 0, ) be coiuouly differeible fucio uch h f > 0 d lim f. The f i id o be low f icreig fucio (.i.f. i hor) if lim 0 f Wrie F { f f } : i.i.f.... Emple (i) f log, > i.i.f. Noe h lim f lim log d f, > d f i coiuou f lim lim 0 f log (ii) f loglog, > e i lo.i.f.. SOME PROPERTIES.. Theorem Le f, g F d le > 0, c > 0 be wo co he we hve (i) f + c (ii) f c (iii) cf (iv) fg (v) f (vi) f ο g (vii) log f (viii) f + g ll lie i F. Give h f, g F d > 0, c > 0 be co. of (i), (ii), (iii), d (iv) follow he defiiio. (v) Le h f Noe h lim h lim f, d f f > 0, d i coiuou f f lim h () () () () lim Hece h f F (vi) Le h fοg ieh. f( g) f lim 0. f Noe h lim h lim f( g ), d () f (()) g g () > 0, d i coiuou f ( g) g g f ( g) g lim lim lim 0. h f( g ) f( g ) g Hece h fο g F (vii) Le h log f Noe h lim h lim log f, d f > 0, d i coiuou f f f f lim lim lim 0. h log f f log f Hece h log f F (viii) Le h f + g For ufficiely lrge, we hve 0 0 g g f + g g f f d f + g f 480

2 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. By ddig he bove, we ge lim 0 Hece h f + g F h () f g 0 lim lim + lim 0 h () f g.. Theorem Le f, g F. Defie h f ( ) d k f( g ) for ech, Give h f, g F. Defie h f ( ) d k f( g ) for ech. Le h () f( ) Noe h lim h lim f( ), d f ( ) > 0, d i coiuou f ( ) f ( ) lim lim lim 0 h f( ) f( ) Hece h f( ) i.i.f. Le k f( g ) Noe h lim k lim f( g ), d k f ( g) g + g > 0 d k i coiuou f ( g) g + g lim lim h f( g ) g f ( g) g f ( g) g lim + lim 0 f( g ) f( g ) g Therefore k f( g ) i.i.f. Hece hk, F.3. Theorem Le f, g F be uch h f d f f lim d 0. g d > g The F. g Give h f d f f, g F,lim d 0 g d > g f ( g ) f( g ) Le f h d g g f g f g g f g lim lim lim lim 0 h f f g g Hece f F g.4. Theorem Le h :, ) ( 0, ) be coiuouly differeible fucio uch h > 0 d lim h (i) Defie g h(log ). The g F lim 0 h (ii) Defie e. The k F lim 0 h k Give h > 0 d lim h (log ) (i) Defie g h(log ) he g Suppoe g F he g ifie g (log ) i.e. lim 0 h(log ) lim 0 g (log ) lim 0 h(log ) () Pu log o h lim 0 h () i.e. lim 0. h Coverely uppoe lim 0 h Pu e o h log d (log ) lim lim 0 h h(log ) g () (log ) Now lim lim lim 0. g () h(log ) h Hece g F (ii) Like proof of (i).5. Theorem If f F he log f lim 0. log 48

3 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. Give h f F, f log f f lim lim log f ' log f lim 0 i.e. lim 0 f log (byl Hopil rule)..6. Theorem f F if d oly if o ech > 0 here d f ei uch h 0, d < > We hve d f f f f f + d f f ' Suppoe f F he lim 0 f i.e. For ech > 0 here ei uch h > Ad hold. f ' d 0 <, > 0, d < > To prove he covere ume h he codiio Le > 0 be give. The here ei uch h > d f We hve, by hypohei 0 d < hi implie h f ' 0 <, > f f ie.. 0 f Therefore f F..7. Theorem If f F he f ' lim 0. f lim 0, for ll > 0 For y wih 0 < <, we ge by Theorem.6, d f 0, d < Thi implie h Hece for ll > for ome f i decreig for > f bouded bove, y, by M Th i, here ei M > 0 uch h f 0 < < M, > f f lim lim 0.8. Noe We kow h ech f F i icreig fucio., by he bove heorem, i i cler h f lim 0, >0. Thi how h he icreig ure of f i low. I oher word, f doe o icree rpidly. Thi juifie he me give o he member of F. From he bove heorem, we hve he followig reul:.9. Theorem If f F lim f 0. he f lim 0 d f I Theorem.7 pu, oge lim 0. f If f F, he lim 0 f f Sice lim 0 we mu hve lim f Theorem Le f F, he for y > d erie f diverge o. + We wrie f ( f ) we kow h he erie diverge o, he 48

4 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. Give > + > 0 If 0 + he lim f If > 0 he + lim lim (from Theorem.7) f f i.e. f diverge o A impor byproduc of he bove heorem i he followig reul... Theorem Le f F. The for y d lim. + f + > d, From Theorem.0, we hve + lim f + lim f, >, + From Theorem.0, we hve lim f d f () d Coider lim + f + f () lim + f () + f () f () + f lim f f + + f (By L Hopil rule) f (i) If lim, he f i id o ympoiclly g equivle o g. We decribe hi by wriig f g. (ii) f Ο( g) Me f Ag for ome A > 0. I hi ce we y h f i of lrge order g. f (ii) f ο( g) Me lim 0. I hi ce we g y h f i of mll order g..3. Emple (i) Coider f, g +, for ll > 0 f d lim lim g + Therefore f g. (ii) Ο (0 ) Becue (0 ) (iii) + + ο( ) Becue lim 0. A reul of Theorem., we ge he followig reul priculr ce..4. Theorem Le f F. The we hve he followig eme. (i) f d f (ii) f ( d ) f (iii) Le f F (i) Pu 0 i Theorem., we ge f () d lim f d f () f f () d f (ii) Pu 0, i Theorem., we ge.. Defiiio Le f, g: [, ) ( 0, ) 483

5 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. f () d lim f f () d f (iii) Pu 0, - i Theorem., we ge d lim d f () f f.5. Theorem Le f F. The f( + (i) lim, f (ii) If f i decreig he c Le f F For y c f( lim, for y f (i) Ce (). Suppoe c > 0 By Lgrge me vlue heorem, There ei, + c uch h f ( + f ( + c ) f f ( + f cf () 0 f f f ( + cf () 0 lim lim, (, + f () f ( + lim 0, ice lim f 0 (by Theorem.9) f () f ( + lim. f Ce (b). Suppoe c < 0 By Lgrge me vlue heorem here + c, uch h ei f f + c ( f f f + c cf () 0 f f f ( + f () 0 lim clim + c, f (), ( + ) fc lim 0, ice lim f 0 f () (by Theorem.9) f ( + lim. f (ii) Ce (). Suppoec > By Lgrge me vlue heorem here, c uch h ei f ( f ( c ) f f ( f ( c ) f 0 f f f ( f f () 0 lim ( c ) lim f f, (, Ad f ( ) i decreig f > f fc There forelim 0, ice lim f 0 (by Theorem.9) f () f c lim. f Ce (b). Suppoe c < By Lgrge me vlue heorem here c, uch h ei f f c ( f f f ( ( cf ) 0 f f f f ( f () 0 lim ( lim, f f Ad f ( ) i decreig f > f f ( There fore lim 0, f ice lim f 0 (by Theorem.9) ( c, ) f c lim. f.6. Theorem Suppoe f F i uch h f i decreig. If 0 < c c d g i fucio uch h f( g ) c g che lim. f 484

6 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. Suppoe f F i uch h f i decreig If 0 < c g c f ( c ) f( g ) f( c ) ice f i decreig f ( c ) f( g ) f( c ) f f f f ( c ) f( g ) f( c ) lim lim lim f f f f( g ) lim f (By Theorem.5) f( g ) lim. f 3. APPLICATIONS OF SLOW INCREASING FUNCTIONS TO SOME PROBLEMS OF NUMBER THEORY Thi ecio deil ome pplicio i problem periig o umber heory. We begi wih he followig impor defiiio. 3.. Defiiio Le f F. Through ou ( ) deoe ricly icreig equece of poiive ieger uch h > d lim for ome. f i.e. f( ) There ei everl uch equece. For emple p, he equece of prime umber i icreig order, f log d. p By prime umber heorem we hve lim log 3.. Theorem Le f :(, ) (, ) be.i.f. ( > ) f () d lim d ( b) <. Suppoe ( ) be b he equece of poiive ieger uch h f. () The.... lim. e Give h f :(, ) (, ) be.i.f. f () b Ad f > d lim d ( < b) log log + log f + ο() If i poiive ieger i iervl [, ) The log k log k+ log f( k) + ο() () k k k k Now Sice log i icreig d poiive i (, ) log k log d+ο(log ) k log +Ο (log ) log + ο (3) O he oher hd if 0 he iequliy ο() < ε Therefore for >, we hve ο() ο() k k ε( + ) < ε i.e. ο() ο k ε > we hve for ll (4) We fid h log f ( k) log f d+ο(log f) k f log f d+ο(log f) (5) f We kow h log f f lim lim (By L Hopil rule) f ( ) f lim 0. f 485

7 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. Ο (log f ) ο (6) Now f () d () f f lim lim 0 (By L Hopil rule) f f d ο( ) (7) f From (5), (6) d (7), we ge log f ( k) log f + ο (8) k From (), (3), (4) d (7), we ge k log ( log + ο) + log f + ο + ο k log k log + log f + ο (9) Bu k k log log + log log k log ep... ep log k k k log log log + log + ο() (By 9) k k ep ep k ( f ο ) ( f ο ) e ( f +ο ) ep log + log + () ep log + () ep log () e f e e Therefore... e... lim. e... e I view of he bove heorem d prime umber heorem implie he followig Theorem Le p be he equece of prime umber. The pp... p lim. p e d. I heorem 3. pu p, f log Le k, c be he equece of ieger which hve i heir prime fcorizio k prime fcor. Rfel Jkimczuk [ 4 ] proved h c ( k )! log ( log log ) k, ( k ) A reul of previou heorem, we hve he followig reul Theorem c, k. c, k... c, k lim. c e k, I Theorem 3. pu c,, f ( k )!log d. CONCLUSIONS We pply he reul dicued i hi ricle o look io ome of he pplicio i umber heory. ACKNOWLEDGEMENTS The uhor like o epre heir griude owrd he mgeme of Vrdhm College of Egieerig for heir coiuou uppor d ecourgeme durig hi work. Furher uhor would like o hk he oymou referee for goig hrough he ricle wih fie ooh comb d mkig criicl comme o he origil verio of hi mucrip. k 486

8 VOL. 8, NO. 7, JULY 03 ISSN ARPN Jourl of Egieerig d Applied Sciece Ai Reerch Publihig Nework (ARPN). All righ reerved. REFERENCES [] G. H. Hrdy d E. M. Wrigh A Iroducio o he Theory of Number. Fourh Ediio. [] R. Jkimczuk. 00. Fucio of low icree d ieger equece. Jourl of Ieger Sequece. 3, Aricle 0... [3] R. Jkimczuk A oe o um of power which hve fied umber of prime fcor. J. Iequl. Pure Appl. Mh. 6: 5-0. [4] R. Jkimczuk The rio bewee he verge fcor i produc d he l fcor, mhemicl ciece: Qurerly Jourl. : [5] Y. Shg. 0. O limi for he produc of power of prime. Sci. Mg. 7: [6] J. Rey Por, P. Pi Cllej d C. Trejo A lii Mem rico, Volume I, Ocv Ediio, Edioril Kpeluz