Congruences for sequences similar to Euler numbers

Transcription

1 Coguece fo equece iila to Eule ube Zhi-Hog Su School of Matheatical Sciece, Huaiyi Noal Uiveity, Huaia, Jiagu 00, Peole Reublic of Chia Received July 00 Revied 5 Augut 0 Couicated by David Go Abtact a E a a 0,,,..., Fo a 0 we defie {E a } by [/] whee [/] / o / accodig a o. I the ae we etablih odulo ie owe, ad how that thee i a et X ad a a T : X X uch that E a i the ube of fixed oit of T. ay coguece fo E a MSC: Piay B68, Secoday A07 Keywod: Eule olyoial, equece, coguece, -egula fuctio. Itoductio The Eule ube {E } ad Eule olyoial {E x} ae defied by. e t e t + 0 t E! t < π ad e xt e t + E x t t < π, 0! E-ail adde: zhihogu@yahoo.co URL: htt:// The autho i uoted by the Natioal Natual Sciece Foudatio of Chia No

2 which ae equivalet to ee [6]. E 0, E 0, ad E x + E 0 E x x 0. Eule ube {E } i a iotat equece of itege ad it ha ay oetie ad alicatio. Fo exale, accodig to [] we have E / h 4 od, whee i a ie of the fo 4 + ad hd i the cla ube of the fo cla gou coitig of clae of iitive, itegal biay quadatic fo of diciiat d. I 005, Aia de Reya[] howed that thee i a et X ad a a T : X X uch that E i the ube of fixed oit of T. I [] the autho itoduced the equece S 4 E 4 ad howed that h 8 S od fo ay odd ie. I [4] the autho yteatically tudied the equece U E. Iied by the oetie of {E }, {S } ad {U }, we ty to itoduce oe equece of itege iila to Eule ube. Fo thi uoe, we itoduce the equece {E a } fo a 0 give by [/] a E a a 0,,,..., whee [x] i the geatet itege ot exceedig x. Actually, E a a E a, E E, E S ad {E a } i a equece of itege. I the ae we aily tudy the oetie of E a. We how that thee i a et X ad a a T : X X uch that E a i the ube of fixed oit of T. Thi geealize Aia de Reya eult fo Eule ube. I Sectio we etablih oe coguece fo E a odulo a ie. Fo exale, fo a ie > we have E / 0,h 4 o h od accodig a 5 od, od o od 4. Let Z ad N be the et of itege ad oitive itege, eectively. I Sectio we etablih oe geeal coguece fo E a +b odulo, whee a Z,,, N ad b {0,,,...}. Fo exale, we deteie E a +b od +4+t, whee t i the oegative itege give by t a ad t+ a. I the cae a, the coguece wa give i []. The coguece ca be viewed a a geealizatio of the Ste coguece [8,6] E +b E b od + fo eve b. Fo N let Z be the et of atioal ube whoe deoiato i coie to. Fo a ie, i [0] the autho itoduced the otio of -egula fuctio.

3 If f Z fo 0,,,... ad f 0 od fo all N, the f i called a -egula fuctio. If f ad g ae -egula fuctio, fo [0, Theoe.] we ow that f g i alo a -egula fuctio. Let be a odd ie, ad let b be a oegative itege. I Sectio 4 b+[ 4 ] +b E +b ad f +b ae -egula fuctio, whee a i the Ja- Uig the oetie of -egula fuctio i [0,], we deduce od. Fo exale, fo, N we have E ϕ +b b+[ 4 ] b E b od, whee ϕ i Eule totiet fuctio. I additio to the above otatio, we alo ue thoughout thi ae the followig we how that f b+ +b E [ + ] 6 cobi ybol. ay coguece fo E ad E otatio: {x} the factioal at of x, od the oegative itege α uch that α but α+ that i α, µ the Möbiu fuctio.. Coguece fo E a odulo a ie Defiitio.. Fo a 0 we defie {E a } by [/] a E a a 0,,,... By the defiitio we have E a Z fo a Z ad E E. The fit few Eule ube ae how below: E 0, E, E 4 5, E 6 6, E 8 85, E 0 505, E 70765, E , E The fit few value of E ad E ae give below: E 0, E, E, E, E 4 57, E 5 6, E 6 76, E 7 46, E , E , E ; E 0, E, E 5, E 46, E 4 05, E 5 6, E 6 65, E , E , E The Beoulli ube {B } ad Beoulli olyoial {B x} ae defied by B 0, B 0 ad B x B x 0.

4 It i well ow that ee [6]. E x x E x B + x + B + + I aticula,. E E It i alo ow that ee [6] + + B + x + ad E 0 + B +. + x B +.. B + 0, B x B x ad E x E x. Thu, Theoe.. Let be a oegative itege ad a 0. The E a a E a Poof. By Defiitio. we have e at 0 [/] a a E + + B + + a. a t! 0 t a! 0 [/] E a a E a t! t eat + e at! 0 E a t.!.4 0 E a t! e at e at + e at / et e at +. Fo. we ow that 0 E a at we deduce E a a E a By. ad. we have + B + at 0 +!! et e at + a a E 4 0. Hece, fo the above ad. [/] a a E. E 0 at! e at +.

5 Thu 0 E a t! et e at + t 0! + + a B + t +! + + a B + t +! 0 ad o E a + + a B + +. The oof i ow colete. Coollay.. Let a 0 ad N. The E a { 0 if, + + a B + + if. Poof. By Theoe. ad the bioial iveio foula we have E a + + a B + +. Notig that B + 0 fo eve we deduce the eult. Lea.. Fo N we have E + E. Poof. Uig.,. ad.4 we ee that E 0 t! 0 E t + t et! e 6t + + e t e 6t + et + e 5t e 6t + et e t + + et e 6t + + t E 0! + t E 0!. So the eult follow. I [], Evall howed that fo a ie od 4,.5 E / h 4 od ad o E /. I [] the autho defied {S } by S S 0 ad howed that S 4 E 4. Thu, by Theoe. we have S E. Fo [, Theoe. ad Coollay.] we ow that fo ay odd ie,.6 h 8 E / od ad hece E /. Now we tate the iila coguece fo E / od. Theoe.. Let be a ie geate tha. The 0 od if 5 od, E E / h 4 od if od, h od if od 4. 5

6 Poof. If od 4, by Lea. ad.5 we have E + E + E { 0 od if 5 od, E h 4 od if od. Now aue od 4. It i ow that ee [6] B 6 B. Thu, B + 6 B + { 0 od if 7,, od 4, B + od if 9 od 4. Hece, by Theoe. ad. we have E 6 E B + 8B + 8B + B B + B + + / B + B + 6 od if 7 od 4, od if, od 4, + 8B + od if 9 od 4. Now alyig [, Theoe.ii] we obtai E theoe i oved. / h od. So the Rea. I a iila way, oe ca how that fo ay ie,9 od 0, h 5 E 5 od. Coollay.. Let be a odd ie with 5 od. The E /. Poof. Fo od 4, it i well ow that [5,.-5] h <. So the eult follow fo Theoe. ad.5. Theoe.. Let be a odd ie, {,,4,...} ad ± od. The <i< i i [ ] E/ od 6

7 ad [/] i i [/] i E/ od fo,4,...,. Poof. Let {} {,4,..., }. Puttig ad ubtitutig by i [, Coollay.] we ee that { E 0 [ } ] E [/] [/] i i i od. i i It i well ow that [5] B od. Thu, i view of. ad. we have E 0 B 0 od if, B od if. Uig. ad Theoe. we ee that { } E E E / Fo the above we deduce [/] i i E / od if, E E E/ od if +. i i ± [/] E/ od. Taig we have the ow eult [/] i i i od. Hece <i< i [/] i i [/] i i i [/] E/ od. i Fo {,4,..., }, uig. ad Theoe. we ee that E { i } E E / E / od. Now uttig all the above togethe we deduce the eult. 7

8 . Coguece fo E a odulo I [] the autho etablihed ay coguece fo E od, whee, N. I the ectio we exted uch coguece to E a +b od, whee a i a ozeo itege ad b {0,,,...}. Lea.. Let ad be oegative itege. The i if, 0 if <. ii if, + 0 if <. Poof. i ca be foud i [4,.64]. We ow ue i to deduce ii. By i we have if +, if, if <. Thi ove ii. Theoe..Let a be a ozeo itege, N ad let b be a oegative itege. Suoe that α N i give by α < α. i If i a odd ie divio of a, the ii We have { E a 0 od od a +b if, 0 od +oda if. { E a 0 od +od a α +od + if, 0 od od a+ α if 8

9 ad iii We have { E a 0 od +od a+ + if, 0 od od a+ od + if. Moeove, if ad b, the ad E a +b 0 od +od a α. E a +b 0 od +od a+ α. Poof. Uig Theoe., Lea. ad. we ee that E a 0 E a a + B a + B a + B a + B a + B Fo Coollay. we ee that fo odd,. + + B a + B a + B + + a + B E Z. + +.

10 Thu, if i a odd ie with a, the + a + B od oda. Now, fo the above we deduce that fo i 0,,. E a +i { 0 od od a if, 0 od +od a if. Fo [0,.5] we ow that fo ay fuctio f,. f Thu,.4 E a +b [b/] f. E a +[ b ]+b [ b ] [ b ] + + E a +b [ b ]. Now alyig. we deduce i. Suoe {, +,..., } ad. If, the ad o 0 od od. Sice B + od ad od+ + < α+, we ee that od + α ad o If, we ee that Theefoe, od a + B + + od a + B + + E a Sice B + od fo odd we alo have E a + + od a α + od + od a α + od. a od od a α. + a + B + + { 0 od +od a α +od + if, 0 od od a+ α if. + + a + B + + { 0 od +od a+ if, 0 od od a+ od + if. 0

11 So ii hold. Sice od+ + α we ee that od + α. Thu, fo ii we deduce.5 E a +i 0 od +od a α fo i 0,. A α + α o α +, we ee that + α + α ad hece α α fo. Fo 0, by.5 we have + + E a +b [ b ] 0 od + α +++od a. Sice + α + α, we ut have + + Cobiig thi with.4 we obtai E a +b [ b ] 0 od α +od a. E a +b 0 od +od a α ad o Now we aue ad b. Fo, N we have + > 7 5 < 6. Hece log + + log + log < 4 ad o + + od a log + + > + + od a log +. Sice od ad + α we ee that od + + log + + ad log + α. Thu, fo odd we have + + od a od od a log + + ad o by ii od a log od a log od a α E a + 0 od ++od a α.

12 Fo eve, uig ii ad the fact + + od a od a od a α we ee that.6 i alo tue. Thu alyig.4 we deduce that E a +b 0 od ++od a α. Thi colete the oof. Coollay.. Let a be a ozeo itege ad b {0,,,...}. The f E a +b i a egula fuctio. Poof. Let α N be give by α < α. A >, we ee that α ad o α. Now alyig Theoe.iii we obtai the eult. Theoe.. Suoe that a i a ozeo itege,,,,t N ad b {0,,,...}. Fo N let α N be give by α < α ad let e a,b E a ]. The E a t+b Moeove, E a t+b +b [ b E a t+b od ++od a α. E a t+b + t e a,b od ++od a+ α +. I aticula, whe ad b, we have E a t+b E a t+b od ++od a+ α +. Poof. Fo N et A a,b 0 E a +b. A α, uig Theoe.iii we ee that A a,b Z ad { 0 od +od a α ++od a if ad b,.7 A a,b 0 od +od a α othewie. By [9, Lea.] we have.8 E a t+b E a t+b. 0 E a t+b

13 Fo Coollay. ad the oof of [, Theoe 4.] we ow that E a t+b.9 0 A a,bt t + A a,b t + + j+ j, j j! j! whee {,} ad {S,} ae Stilig ube give by ad xx x + x,x S,xx x +. By [, Lea 4.], fo + j we have.0, j j! j,! S j,! j! j Z ad,!!,!! S j,! j t j, j! od. A α + α + we have + α + α ad hece α α fo. Theefoe, by.7 we have +od a++ α + A a,b fo +. Hece, uig.9 we get. E a t+b t A a,b od ++od a+ α +. Fo.7 we have +od a α A a,b. Sice α + α o α + we ee that od a α od a α. Hece, by. we get. E a t+b 0 od ++od a α. Fo + we have α + α + ad o + + od a α + od a α +. Thu, uig. we ee that fo +,. E a t+b 0 od +++od a+ α +.

14 Whe ad b, by Theoe.iii ad the fact α + α we have od A a,b + od a + α + + od a + α +. Thu, it follow fo. that.4 By.4 we have.5 A a,b [b/] Fo Theoe.iii we ow that E a +b 0 od ++od a+ α +. [b/] E a +b [ b ]. E a +b [ b ] 0 od +od a+ α +. Fo N we have + od a + α + + od a α +. Thu, fo the above we deduce that.6 A a,b e a,b od +od a++ α +. Now cobiig.-.4,.6 with.8 we deive the eult. Theoe.. Let a be a ozeo itege,, N, ad b {0,,,...}. The E a +b Ea b a b + 5 a + a b od +4+oda if a, ab b + od +4 if a ad b, a od +4 if ab. Poof. Fo N let α Z be give by α < α, ad let A a,b E a +b ad e a,b E a +b [ b ]. Sice,! ad S j,, taig ad t i.9 we ee that.7 { E a b E a +b A a,b + j A a,b j, j j! j j j j }.! j! 4

15 Fo j it i eaily ee that j j! j 0 od 4. By Theoe.iii we have +od a α A a,b. Thu, fo 5 we have od A a,b + od a α 5od a + 5 α 5 + 5od a. Set H Fo the defiitio of Stilig ube we ow that,!h fo. Thu, fo,,! H.!! Hece, fo the above we deduce that E a +b Ea b Set A a,b + A a,b H od +4+oda. f H α H α. Sice α H Z we ee that f 0 od 4 fo 5. It i eaily ee that f, f ad f 4. Hece, fo the above we deduce that.8 { } E a +b Ea b + A a,b + A a,b 0 od +4+oda if ad, { 4 A a,b + A 4 a,b} od +4+oda if > o >. Fo.6 we ee that.9 A 4 a,b e 4 a,b od +5oda ad A a,b e a,b od +4oda. If b, by Theoe.iii we have A 4 a,b 4 4 Fo Theoe. we have 4 E a +b 0 od +5od a..0 E a 0, E a a, E a a, E a a + a, E a 4 4a + 8a, E a 5 5a + 0a 6a 5, E a 6 6a + 40a 96a 5, E a 7 7a + 70a 6a 5 + 7a 7, E a 8 8a + a 896a a 7. 5

16 Hece, if b, fo.9 ad.0 we deduce that A 4 a,b e 4 a,b E a 0 4E a + 6E a 4 4E a 6 + E a 8 6 8a 5 7a 4 0 od +5oda. Theefoe, we alway have A 4 a,b 0 od +5oda. Fo.0 we ee that. e a,b 8 Ea 0 E a + E a 4 E a 6 a 6a if b, E a + E a 5 E a 7 a 7a 4 + 8a if b. 8 Ea Thu, alyig.9 we get A a,b e a,b + ab a od +od a. Theefoe, 4 A a,b 4 + ab a 4a + ab + od 4+od a. Hece, by the above ad.8 we obtai. { E a +b Ea b A a,b + A a,b } 4a + ab + od +4+oda. Fo Theoe.ii we ee that. 0 E a +b [ b ] 0 od 7+5od a fo 4. Thu, by.5 we have A a,b [b/] [b/] E a 0 [ b +b [ b ] 4e a,b [ b 4 Fo.0 we ow that.4 e a,b 4 E a +b [ b ] ] 0 ] 8e a,b e a,b [ b E a +b [ b ] ] e a,b od 4+od a. { a E b [ b ] Ea + a if b, +b [ b ] Ea 4+b [ b ] 4a a if b. Thi togethe with. yield e a,b [ b] e a,b a b a 6a a + b 6a b if b, 4a a b a 7a 4 + 8a a 7a 4 b + 6 8ba + b + if b. 6

17 Thu,.5 A a,b e a,b [ b] e a,b a + b od 4+oda if a, a b od 4+oda if a ad b, 0 od 4+oda if ab. By. ad.5 we have A a,b [b/] [b/] + + E a 0 +b [ b ] [b/] + + E a 0 +b [ b ] e a,b [ b] [ b 4e a,b + ] 8e a,b e a,b [ b] e a,b + [ b][b] e a,b od 4+oda. Fo.0 we have.6 e a,b { a E a if b, b [ b ] Ea +b [ b ] a a if b Hece, fo the above we deduce.7 A a,b e a,b [ b] e a,b + [ b][b] e a,b a b a bb + a 6a a a b + 6a 4 bb a a b od 4+oda if b, a a b 4a a + b b a 7a 4 + 8a a a b od 4+oda if a ad b, a a b 4a a + b b a 7a 4 + 8a a od 4+oda if a ad b. Now ubtitutig.5 ad.7 ito. we obtai the eult. Fo a, b ad 4, by Theoe. we have { Eb + 5 od +4 if b 0,6 od 8, E +b E b od +4 if b,4 od 8. 7

18 Thi ha bee give by the autho i []. Coollay.. Let be a oegative itege. The { E 56 + od 5 if 4, od 5 if 4, { E 00 od 5 if 4, od 5 if 4, E 4+ 5 od 5, E od 5. Poof. Taig a ad i Theoe. we deduce the eult fo. Sice E 0, E, E ad E, we ee that the eult i alo tue fo 0. Theoe.4. Let a be a ozeo itege,, N ad b {0,,,...}. The E a +b Ea +b Ea b od ++oda. Poof. Fo N et e a,b E a +b [ b ]. Fo.4 we ow that a e a,b ad o +oda e a,b. Now taig ad t i Theoe. ad the alyig the above we deduce the eult. 4. Coguece fo E +b ad E +b od Let be a odd ie. I [] the autho howed that f +b E +b i a -egula fuctio whe b i eve. I thi ectio we etablih iila eult fo E ad E, ad the ue the to deduce coguece fo E +b ad E +b od. Lea 4.. Let N, {0,,,..., } ad b {0,,,...}. Let be a odd ie ot dividig. The f +b E +b A A E +b +b i a -egula fuctio, whee A {0,,..., } i give by A od. Poof. Fo x Z let x be the leat oegative eidue of x odulo. Fo [0,Theoe.] we ow that B +b+ x B +b+ + b + B +b+ x+ x +b B +b+ + b + 8

19 i a -egula fuctio. Hece g B +b+ + B +b+ + b + +b B +b B +b+ + b + + i a -egula fuctio. Let A {0,,..., } be uch that A od. The A ± + od ad A A o A accodig a A < o A. A + + A A ad uig. we ee that B +b A A + B +b+ + b + + A + A if A <, if A, B +b+ A + B +b+ A +b+ E + b + +b A if A <, B +b+ A B +b+ A +b+ E + b + +b A +b+ A A E +b. Alo, by. we have if A. B +b+ + B +b+ +b+ E + b + +b. Thu, fo the above we ee that g +b+ E +b A A E +b +b 9

20 i a -egula fuctio. By Feat little theoe we have +b b 0 od. Thu +b+ +b i a -egula fuctio. Hece, by the oduct theoe fo -egula fuctio [0, Theoe.], f +b+ +b g i a -egula fuctio a aeted. Fo Lea 4. we have the followig eult. Lea 4.. Let be a odd ie, {,,4,...} ad ± od. Let b be a oegative itege ad {,,..., }. The f +b +b E +b if od, + b++ +b +b E +b if od i a -egula fuctio. Poof. Let A {,,..., } be uch that A od. The clealy A o accodig a o od. Sice E x E x, we have E +b +b E +b b E +b. Now alyig the above ad Lea 4. we deduce the eult. Theoe 4.. Let be a odd ie ad let b be a oegative itege. The i f b+[ 4 ] +b E +b i a -egula fuctio. ii f [ + ] 6 b+ +b E +b i a -egula fuctio. Poof. Puttig 4 ad i Lea 4. ad alyig Theoe. we obtai i. Puttig 6 ad i Lea 4. ad alyig Theoe. we obtai ii i the cae >. Fo Theoe.i we ee that ii i alo tue fo. So the theoe i oved. Fo Theoe 4. ad [, Theoe 4. with t ad d 0] we deduce the followig eult. Theoe 4.. Let be a odd ie ad,, N. Let b be a oegative itege. The b+[ 4 ] ϕ +b E ad [ + 6 ] ϕ +b b+ ϕ +b E ϕ +b 0 b+[ 4 ] ϕ +b E ϕ +b od

21 [ + 6 ] b+ ϕ +b E ϕ +b od. I aticula, fo we have E ϕ +b b+[ 4 ] b E b od ad E + ϕ +b [ ] 6 b+ b E b od. Lea 4.. See [0, Theoe.]. Let be a ie, N ad let f be a -egula fuctio. The thee ae itege a 0,a,...,a uch that f a + + a + a 0 od fo 0,,,... Moeove, if, the a 0,a,...,a od ae uiquely deteied ad od! a fo 0,,...,. Fo Theoe 4. ad Lea 4. we deduce the followig eult. Theoe 4.. Let be a odd ie, N ad. Let b be a oegative itege. The thee ae uique itege a 0,a,...,a,c 0,c,...,c {0,±,±,..., ± } uch that fo evey oegative itege, b+[ 4 ] +b E +b a + + a + a 0 od ad [ + 6 ] b+ +b E +b c + + c + c 0 od. Moeove, od! a ad od! c fo 0,,...,. Coollay 4.. Let N. The i E od 7, E od 7; ii E od 5 ; iii E od 5; iv E od 5; v E od 5. Poof. A E 0, E ad E, taig i Theoe 4. we ee that E od 7 ad + + E od 7. Thi yield i. Pat ii-v ca be oved iilaly. Coollay 4.. Let N. The i E 6 + od 7, E + 6 od 7; ii E od 5; iii E od 5; iv E od 5; v E od 5.

22 5. { E a } i ealizable Let {b } be a give equece of itege, ad let {a } be defied by a b ad a b + a b + + a b,,4,... If {a } i alo a equece of itege, followig [] we ay that {b } i a Newto-Eule equece. Lea 5.. See [4, Lea 5.]. Let {b } be a equece of itege. The the followig tateet ae equivalet: i {b } i a Newto-Eule equece. ii d µ d bd 0 od fo evey N. iii Fo ay ie ad α, N with we have b α b α od α. iv Fo ay,t N ad ie with t we have b b od t. v Thee exit a equece {c } of itege uch that b d dc d fo ay N. Poof. Fo [, Theoe ] o [] we ow that i, ii ad iii ae equivalet. Clealy iii i equivalet to iv. Uig Möbiu iveio foula we ee that ii ad v ae equivalet. So the lea i oved. Let {b } be a equece of oegative itege. If thee i a et X ad a a T : X X uch that b i the ube of fixed oit of T, followig [7] ad [] we ay that {b } i ealizable. I [7], Pui ad Wad oved that a equece {b } of oegative itege i ealizable if ad oly if fo ay N, d µ d b d i a oegative itege. Thu, uig Möbiu iveio foula we ee that a equece {b } i ealizable if ad oly if thee exit a equece {c } of oegative itege uch that b d dc d fo ay N. I [] J. Aia de Reya howed that {E } i a Newto-Eule equece ad { E } i ealizable. Lea 5.. See [6,.0]. Fo N ad 0 x we have E x 4! i + πx π π Taig x 4 i Lea 5. ad alyig Theoe. we deduce 5. 0 [ ] + + E π +.! 4 Theoe 5.. Let a, N. The E a > 4+ a! π + a + + > 0

23 ad E a E a < 4+ a! π + a Poof. By Theoe. ad Lea 5. we have a E a 4+ a! π + 4+ a! π + 0 a 4a 4! π + i +π a + + < 4+ a! π a! π + + π i a Fo {0,,...,a } we have i +π a Thu, i i +π a π a + a +. i +π a 4a a a + a > 0 ad o + π a 4a > 0. 4a + a E a > 4+ a! π + a + + i π a. It i well ow that ix π x fo 0 x π. Thu i π a a. So the fit iequality i tue. Sice a a < a + π i a 4a a + a a a + 4a < + a +, cobiig the above we obtai the eaiig iequality. Theoe 5.. Let N with. The + E > 0 ad + E > 0. Poof. Fo 0 we ee that

24 Thu, > > [ ] Now alyig 5. we deduce + E > 0. Siilaly, fo 0 we have > Thu, uig Lea 5. ad Theoe. we obtai + + E π + 4 6! E 6 π + 4! + i + π > > 0. Hece, + E > 0. The oof i ow colete. Theoe 5.. Let a be a oitive itege. Fo ay ie divio of N we have E a Ea / od od. Hece {E a } i a Newto-Eule equece. Poof. Suoe ad 0 with 0. Fo Theoe. we ee that E a Ea 0 od fo ad N. It i well ow that E. Thu, uig Theoe. we ee that E a a E 0 + a E od. Hece, E a Ea 0 E a 0 E a od. Now uoe that i a odd ie divio of ad with. If a, by Theoe. ad the fact Z fo we have E a a a + a E a od 4

25 ad / a + E a Sice a, we have a a E a od. a a ϕ + a a od. Thu E a Ea / od. Let u coide the cae a. Suoe that A {0,,...,a } i give by a A. Fo Lea 4. we ow that fo a give oegative itege b, f a +b E +b a A a a +b E +b A a +b i a -egula fuctio. By [0, Coollay.] we have f f 0 od. Thu, uig Theoe. we obtai 5. E a +b Ea b od fo b. A, uig 5. we ee that E a Ea Ea + E a E a / od. Now uttig all the above togethe with Lea 5. we obtai the eult. Theoe 5.4. Let a N. The { E a } i ealizable. Poof. Suoe that i a ie divio of ad t od. Fo Theoe 5. we ow that E a Ea / od t. It i eaily ee that / od t. Thu, E a / E a / od t. Hece, uig Lea 5. we ow that { E a } i a Newto-Eule equece ad o d µ d d E a d Z. By Theoe 5., E a > 0. Now it eai to how that d µ d d E a d 0. Fo.0 we have E a a ad E a 4 4a + 8a. Thu the iequality i tue fo,. Fo ow o we aue. Obeve that + a < a ad + a fo N. Uig Theoe 5. we ee that fo N, 5. Hece a! π + < E a < 4+ a +! π +. µ/d d E a d d 5

26 E a E a + d,d [/] d 6 7 6a π! 4a µ/d d E a d d E a d > a! π + [/] π d 6 7 6a π! { + + 4a 4a d! π π 4a Fo a we have 4a > 4a [ π π ]+ 4a ad π > 7 6 [/] d π 7 6 π 4 4a π /4 6/π 7 6 π 4 4 d+ a d+ d! π d+ π [ ]+ 4a π 4a π 4a /π. Thu, fo the above we deduce d µ d d E a d > 0. Fo a we ee that π > π 6 6 π 4 4 π 4 π [ ]+ 4 π 7 > 4/π 6 ad o d µ d d E a d > 0 by the above. Now uaizig the above we ove the theoe. 4/π > 0 Acowledgeet. The autho tha the efeee fo hi helful coet ad valuable uggetio o iovig Theoe.. Refeece [] J. Aia de Reya, Dyaical zeta fuctio ad Kue coguece, Acta Aith , 9-5. [] B.S. Du, S.S. Huag ad M.C. Li, Geealized Feat, double Feat ad Newto equece, J. Nube Theoy 98 00, 7-8. [] R. Evall A coguece o Eule ube, Ae. Math. Mothly 89 98, 4. [4] H.W. Gould, Cobiatoial Idetitie, A Stadadized Set of Table Litig 500 Bioial Coefficiet Suatio, Rev. Ed., Mogatow Pitig ad Bidig Co., Wet-Vigiia, }.

27 [5] K. Ielad ad M. Roe, A Claical Itoductio to Mode Nube Theoy d editio, Sige, New Yo, 990,.,48. [6] W. Magu, F. Obehettige ad R.P. Soi, Foula ad Theoe fo the Secial Fuctio of Matheatical Phyic d editio, Sige, New Yo, 966,. 5-. [7] Y. Pui ad T. Wad, Aithetic ad gowth of eiodic obit, J. Itege Seq. 400, At. 0.., 8. [8] M.A. Ste, Zu Theoie de Euleche Zahle, J. Reie Agew. Math , [9] Z.H. Su, Coguece fo Beoulli ube ad Beoulli olyoial, Dicete Math , 5-6. [0] Z.H. Su, Coguece coceig Beoulli ube ad Beoulli olyoial, Dicete Al. Math , 9-. [] Z.H. Su, O the oetie of Newto-Eule ai, J. Nube Theoy 4005, 88-. [] Z.H. Su, Coguece ivolvig Beoulli olyoial, Dicete Math , 7-. [] Z.H. Su, Eule ube odulo, Bull. Aut. Math. Soc. 8 00, -. [4] Z.H. Su, Idetitie ad coguece fo a ew equece, It. J. Nube Theoy, to aea. [5] J. Ubaowicz ad K.S. Willia, Coguece fo L-Fuctio, Kluwe Acadeic Publihe, Dodecht, Boto, Lodo, 000. [6] S.S. Wagtaff J., Pie divio of the Beoulli ad Eule ube, i: M.A. Beett et al. Ed., Nube Theoy fo the Milleiu, vol. III Ubaa, IL, 000, A K Pete, 00,