A NOTE ON ENNOLA RELATION. Jae Moon Kim and Jado Ryu* 1. INTRODUCTION

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol 8, No 5, pp 65-66, Ocober 04 DOI: 0650/m Th paper avalable ole a hp://ouralawamahocorw A NOTE ON ENNOLA RELATION Jae Moo Km ad Jado Ryu* Abrac Eola ve a example of a relao amo he cycloomc u whch o a combao of elemeary relao He alo prove ha wce ay relao amo he cycloomc u a coequece of elemeary relao I he ee of he drbuo, he oro par of he uveral eve pucured drbuo A 0 a -oro roup I parcular, whe ha hree dc prme dvor, A 0 ha a uque -oro eleme The am of h paper o fd a alorhm o produce he uque -oro eleme whe ha hree dc odd prme dvor INTRODUCTION For a pove eer mod 4, le ζ e π/ be a prmve h roo of C For a eer k wh k, pu a k lo ζ k, whch he loarhm of a cycloomc umber I well kow ha here are wo ype of relao amo he cycloomc umber: a k a k for k a /mk /m 0 a km for m ad m, k We call hee relao he elemeary relao I [], Eola ve a relao for 05 whch o a combao of elemeary relao: a a a 7 a 4 a 44 a 46 a a 9 a 6 a 5 a 40 a 8 0 Receved Auu 7, 0, acceped February 5, 04 Commucaed by We-Ch We L 00 Mahemac Subec Clafcao: Prmary R8; Secodary R7 Key word ad phrae: Cycloomc u, Eola relao, Drbuo Th reearch wa uppored by Bac Scece Reearch Proram hrouh he Naoal Reearch Foudao of Korea NRF fuded by he Mry of Educao, Scece ad Techoloy 0RAA0059 *Correpod auhor 65

2 654 Jae Moo Km ad Jado Ryu We call uch a relao a Eola relao Le A 0 be he uveral eve pucured drbuo Namely, A 0 he abela roup eeraed by { x x Z/ Z, x } 0 wh he relao: 0 x x for x 0 /m x x m 4 for m ad m x, x m 0 The rucure of A 0 kow o be [4, Theorem 8] A 0 Z ϕ/r Z/Z r r, where r he umber of dc prme dvor of Moreover, he map x/ a x duce a omorphm A 0 / Z/Z r r lo ζ a Thu from he -oro eleme of A 0, we ca oba Eola relao I parcular, A 0 ha a uque -oro eleme whe p e pe pe ha hree dc prme dvor The am of h paper o fd a alorhm o produce a Eola relao whe ha hree dc odd prme dvor Namely, we wll fd he -oro eleme of he uveral eve pucured drbuo Alhouh here aoher alorhm o fd Eola relao [], eem ha our reul more explc ad effce oce he eeraor of Z/p e Z are ve PRELIMINARIES AND NOTATIONS Le p e pe pe be he prme facorzao of whch odd For each, ad, pu q p e, /q ad m ϕq /, whereϕ he Euler-ph fuco We have Z/Z Z/q Z Z/q Z Z/q Z We fx a eeraor of he cyclc roup Z/q Z The uque eer x mod afy x mod q ad x mod alo deoed by Wh hee oao, he relao ad below ca be obaed from he relao ad 4, where p a eer afy p p mod : m m m

3 A Noe o Eola Relao 655 m b b bp for cdb, p I α Throuhou h paper we aume {,, } {α,, } We defe I α ad by I α he dex of p I for he bae α, e, α α p mod q α { Iα f 0 I α <m α, I α We alo defe δ α by I α m α f m α I α <ϕq α { δ α f I α I α, f I α I α Le ad L α L α I α m I α q q p q p q I α m I α q q I q I q I he ummao above ad for he re of h paper, 0 or 0 be uderood o be zero Noe ha L α m 0 I α q q hould ad ha ce L α I α I α I α q p q p 0 q q Lemma For eer α, ad, we have

4 656 Jae Moo Km ad Jado Ryu L α Lα, L α L Proof I o hard o check ha m [ ] τ 0for all τ q q Thu L α L α 0wh τ p We have L α I α I α I 0 I 0 L q I 0 I α I q q q q q I α q q A -TORSION ELEMENT IN THE UNIVERSAL EVEN PUNCTURED DISTRIBUTION Th eco devoed o fd he -oro eleme A 0 Pu We alo defe B α by M M M m 0 m m m m m m m m m k, m m k, k

5 where I B α 0 I B α 0 I B α 0 I αm I α m B α A Noe o Eola Relao 657 B f δ α,δα, B f δ α,δα, B f δ α,δα, B f δ α,δα, m α I α m α I α α I α m B 0 α I α m m I α m I α I α I αm, α, α α m α m m I α I α m m I α, α α Lemma For eer α, ad, we have M α δ α Lα δα Lα B α Proof Fr, we coder he cae whe α Noeha M m 0 m m m m Suppoe ha δ ad δ Thewehave M L L I M p q q k k k p I k k p q q

6 658 Jae Moo Km ad Jado Ryu M I B I m m I For δ δ, wehave k k I m I m I k k M L L M L L L L B L L B ce he mea of M for δ δ ad ha for δ δ δ ad δ, wehave aree Whe M L L M L L M I B I m m k k m ki m km I Fally, for δ ad δ,wehave m k k M L L M L L B m m I ki The cae whe α or ca be mlarly proved by u he dee M m m 0 m m m k kp k kp k ad M m 0 m p

7 A Noe o Eola Relao 659 The Theorem Pu M m 0 R M δ δ he -oro eleme A 0 Proof Oberve ha M M M m 0 m L δ δ m M m 0 m m m k L δ δ L B B B k m O he oher had, by Lemma, we have m m k m m m km k M δ L δ L M δ L δ L M δ L δ L B B B Sce L L, L L ad L L,wehave M δ δ L δ δ L δ δ L B B B 0 Hece R 0 Fally, oe ha R 0ce he coeffce of he expao of R wh repec o he ba of A 0 ve [, Theorem ] equal 4 EXAMPLE Whe 05, he heorem ve he prevou eco eable u o oba he follow Eola relao Le aa for mplcy Pu p q 7, p q 5ad p q The wh mod 05, 4 mod 05 ad 7 mod 05, we have M

8 660 Jae Moo Km ad Jado Ryu Sce δ,δ, δ,δ, δ,δ, we have δ δl 0, δ δl 0, δ δl 0 ad B B , B B 8 0, 6 B B 5 Thu R To compare above relao wh he oe ve by Eola, we oe ha R R R, where R ad R are um of elemeary relao ad 4: R , ad R

9 A Noe o Eola Relao 66 ACKNOWLEDGMENTS We would lke o hak he referee for h/her careful read of he earler vero of h paper ad valuable ueo REFERENCES M Corad, O explc relao bewee cycloomc umber, Aca Arh XCIII, 000, V Eola, O relao bewee cycloomc u, J Number Theory, 4 97, 6-47 R Gold ad J Km, Bae for cycloomc u, Compoo Mah, 7 989, -8 4 L Waho, Iroduco o Cycloomc Feld, Grad Tex Mah, Vol 74, Sprer-Verla, New York/Berl, 980 Jae Moo Km ad Jado Ryu Deparme of Mahemac Iha Uvery Icheo, Korea E-mal: mkm@haackr dryu@haackr

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