ON COVERING EQUIVALENCE. Zhi-Wei Sun

Transcription

1 Aalytic Number Theory Beijig/Kyoto, 999, 77 30, Dev. Math., 6, Kluwer Acad. Publ., Dordrecht, 00. ON COVERING EQUIVALENCE Zhi-Wei Su Abtract. A arithmetic equece a {a+x : x Z} 0 a < with weight λ C i deoted by λ, a,. For two fiite ytem A { λ, a, } k ad B { µ t, b t, } l t of uch triple, if x a λ x b t µ t for all x Z the we ay that A ad B are coverig equivalet. I thi paper we characterize coverig equivalece i variou way, our characterizatio ivolve the -fuctio, the Hurwitz ζ-fuctio, trigoometric fuctio, the greatet iteger fuctio ad Egyptia fractio.. Itroductio ad Prelimiarie By Z + we mea the et of poitive iteger. For a Z ad Z + we let a + Z repreet the arithmetic equece {, a, a, a, a +, a +, } ad write a for a + Z if a R {0,,, }. For a fiite ytem. A {a } k Z + ad a R for,.k of uch arithmetic equece, we defie the coverig fuctio w A : Z N {0,,, } a follow:. w A x { k : x a }. Let m be a poitive iteger. If w A x m for all x Z, the we call. a m- cover of Z; whe w A x m for ay x Z, we ay that A form a exact m-cover of Z. The otio of cover i.e. -cover wa itroduced by P. Erdö [Er] i the early 930, oe of the implet otrivial cover of Z i {0, 03, 4, 56, 7}. Cover of Z have may ice propertie ad iteretig applicatio, for problem Key word ad phrae. Coverig equivalece, uiform fuctio, gamma fuctio, Hurwitz zeta fuctio, trigoometric fuctio. 000 Mathematic Subject Claificatio. Primary B5; Secodary B75, D68, M35, 33B0, 33B5, 33C05, 39B5. The reearch i upported by the Teachig ad Reearch Award Fud for Outtadig Youg Teacher i Higher Educatio Ititutio of MOE, ad the Natioal Natural Sciece Foudatio of P. R. Chia.

2 ZHI-WEI SUN ad reult we recommed the reader to ee R. K. Guy [G], ad Š. Porubký ad J. Schöheim [PS]. Let M be a additive abelia group. I 989 the author [Su] coidered triple of the form λ, a, where λ M, Z + ad a R, we ca view λ, a, a the arithmetic equece a with weight or multiplier λ. Let SM deote the et of all fiite ytem of uch triple. For.3 A { λ, a, } k SM we aociate it with the followig periodic arithmetical map w A : Z M:.4 w A x x a λ for x Z, which i called the coverig map of A. w refer to the zero map ito M. Clearly w A i periodic modulo the leat commo multiple [,, k ] of all the moduli,, k. For A, B SM, if w A w B the we ay that A i coverig equivalet to B ad write A B for thi. Oberve that.3 ad B { µ t, b t, } l t SM are equivalet if ad oly if.5 C { λ, a,,, λ k, a k, k, µ, b, m,, µ l, b l, m l }. We idetify. with the ytem {, a, } k. Thu,. form a exact m-cover if ad oly if A { m, 0, }. Let M be a additive abelia group, ad f a map of two complex variable ito M uch that { x+r, y : r R} Domf for all x, y Domf ad Z+. If.6 x + r f, y fx, y for ay x, y Domf ad Z +, the we call f a uiform map ito M. Note that fx, y f x+0, y ad {r} {0} for ay Z+. I 989 the author [Su] howed the followig Theorem.. Let M be a left R-module where R i a rig with idetity. Wheever two ytem A { λ, a, } k ad B { µ t, b t, } l t i SR are equivalet, for ay uiform map f ito M we have.7 x + a λ f, y l t x + bt µ t f, y for all x, y Domf. A imple proof of thi remarkable theorem wa give i Sectio of [Su8].

3 ON COVERING EQUIVALENCE 3 Let f be a uiform map ito the complex field C with Domf D D ad fx 0, y 0 0 for ome x 0, y 0 D D. If fx, y gxhy for all x D ad y D, the for each Z + we have x0 + r g hy 0 gx 0 hy 0 0, thu θ hy 0 /hy 0 0 ad therefore θhy hy g gx 0 θm hy 0 hmy 0 hmy 0 hmy 0 x0 + r hy for every y D, θmθ for ay m, Z+ ad.8 x + r g θgx for all Z + ad x D. Such fuctio g with D Domg [0, or Q [0, were tudied by H. Walum [W] i 99 where Q deote the ratioal field, ad ivetigated earlier by H. Ba [B], D. S. Kubert [K], S. Lag [La] ad J. Milor [M] i the cae θ where C. I thi paper by R ad R + we mea the field of real umber ad the et of poitive real. For a field F we ue F to deote the multiplicative group of ozero elemet of F. Whe x R, [x] ad {x} deote the itegral part ad the fractioal part of x repectively, if Z + the we write {x} to mea {x/}. For coveiece, we alo et.9 q0 0 ad qz z for z C.. Some Example of Uiform Map For a uiform map f, the map f x, y f x, y i alo a uiform map. Example.. Two imple uiform map ito the additive group Z are the fuctio I, I + : C C Z give below: { if x Z,. Ix, y 0 otherwie; { if x Z +, I + x, y 0 otherwie.

4 4 ZHI-WEI SUN Aother example i the fuctio [ ] : R R Z defied by. [ ]x, y [x], the idetity [ x+r ] [x] for Z+ i well kow ad due to Hermite. Now we tur to uiform map ito the multiplicative group C. Example.. i Defie γ : C R + C a follow:.3 γx, y { xy x / πy if x N {0,,, }, x x! yx πy otherwie. Whe Z + ad y R +, if x C \ N, the by applyig Gau multiplicatio formula z + r π z z with z x/, we obtai that x + r y x+r/ y x x ; πy πy o the other had, for m k + l with k N ad l R, we have lim x m r l lim x m x+r x+r y lim πy x m x+l x + m + m j0 x + j y x / x+m xy x / πy x+l y / πy + k j0 x+l + j y x+l m m! y m πy k k! y k πy. Thu γ i a uiform map ito C. ii A z z π/ i πz for z Z, γx, yγ x, y I +x,y /Sx, y for x C ad y > 0, where { i πx if x Z,.4 Sx, y x y if x Z. It follow that the fuctio S : C C C i a uiform map ito C. iii For α, β, γ C with Reγ > Reα + β ad γ 0,,,, we ue F α, β, γ, z to deote the hypergeometric erie give by F α, β, γ, z + αα + α + ββ + β + z!γγ + γ +

5 ON COVERING EQUIVALENCE 5 which coverge abolutely for z. Let u, v, w C ad D u,v,w coit of all thoe x, y C C uch that x+ w w u y, x+ y, x+ w v y N ad Rex+ w u v y > 0. Whe x, y D u,v,w, by a formula i [Ba] we have F u y, v y, w y + x, x x u/y v/y x u/yx v/y γx, yγx, y γx 3, yγx 4, y 0 where x x + w y, x x + w u v y, x 3 x + w u y Z + clearly x+r F, y D u,v,w for all r R ice x+r u y, v y, w y + x + r, F ad x 4 x + w v y ; for ay + z ad u y, v y, x + r, γ x +r, yγ x +r, y γ x 3+r, yγ x 4+r, y γx, yγx, y γx 3, yγx 4, y F So the fuctio F u,v,w : D u,v,w C give by.5 F u,v,w x, y F u y, v y, w y + x, y x+z/y+r u y, v y, w y + x, i a uiform map ito C. Remark.. Sice the fuctio S i a uiform map ito C, we have log i π x + r log i πx for Z + ad x 0,. I 966 H. Ba [B] howed that every liear relatio over Q amog the umber log i πx with x Q 0,, i a coequece of the lat idetity, together with the fact that log i π x log i πx. See alo V. Eola [E]. Example.3. i Defie G, H : C R + C by.6 Gx, y y ad.7 Hx, y y log y + log y + m0 m qm + x m + + qx m. m We aert that G i a uiform map ito C ad hece o i H G..

6 6 ZHI-WEI SUN Let Z + ad y > 0. By Example.i, x + r y x+r πy xy x πy for x C \ N. Takig the logarithmic derivative of both ide we get that Thu y x + r/ x + r/ + logy x + log y for x 0,,,. x x + r/ x + r/ + logy + γ x y x + log y + γ for x C \ N where γ i the Euler cotat. By a kow formula ee, e.g., [Ba], if x C \ N the ygx, y log y m + x m + x m + m + x x x + γ. m0 So, for ay x C \ N we have m0 x + r G, y Gx, y. Whe x a b where a R ad b N, x + r G x + a G, y, y Gx, y Gx, y + lim z x G b, y Gx, y + lim z x z x lim y z Z + y lim z x y m0 m x m0 m b z Z z Z r a Gz, y G z + r G, y z + a, y qm + x qm + z + q0 q x + z y q m + z + a qm b + q b + z + a q0 y x z + y 0. z + a/ + b

7 Note that for ay v N we have lim u v m0 m v ON COVERING EQUIVALENCE 7 m u m v lim u v mv+ u v m um v 0. For, if u v < / the m u m v u v m um v < / m v / for m v +, v +,, thu the erie mv+ m u m v with u v < / coverge uiformly. ii The Hurwitz zeta fuctio ζ, v i defied by the erie ζ, v m0 m + v for, v C with Re > ad Rev > 0 or v N, by Lemma of [M] it exted to a fuctio which i defied ad holomorphic i both variable for all complex ad for all v i the imply coected regio C\, 0]. I additio, we et ζ, v Gv,. For v C \ N, if Re > the we alo have ζ, v + ζ, v ζ, v + ζ, v If Re > the m m0 m + v + m + m + v m0 m0 m + v v ; m + m + v v. d dv ζ, v m0 dm + v dv ζ +, v for all v C \ N. Wheever C \ {0, }, d ζ, v ζ +, v for all v C \, 0] dv by aalytic cotiuatio. A ζ0, v v, d dv ζ0, v. For thoe v C \ N, we have d d ζ, v dv dv m + ζ, v. m + v m + v m0 m0

8 8 ZHI-WEI SUN For C we defie ζ : C \, 0] R + C by { y ζ, x if,.8 ζ x, y y ζ, x logy if. Let x C \, 0] ad y R +. The ζ x, y Gx, y. If Re > ad Z +, the ζ, x + r k0 ζ, x. x + r + k By part i ad aalytic cotiuatio, for ay C the fuctio ζ i a uiform map ito C. Remark.. a I [M] Milor oberved that there i o cotat c uch that x + r/ x + r/ + c x x + c for Z+ ad x 0,. b By [Ba], if x, y > 0 the dζ x, y d y d 0 d 0 ζ, x y log y ζ, x log x logπ + x log y log γx, y. 0 c For C \ N Milor [M] proved i 983 that the complex vector pace of all thoe cotiuou fuctio g : 0, C which atify.8 with D 0, ad θ, i paed by liearly idepedet fuctio ζ, x ad ζ, x where x rage over 0,. Example.4. i Let ψ be a fuctio ito C with Domψ C. Let D ψ deote the et of thoe x, y C C uch that x + ky Domψ for all k Z ad k a mod ψx + ky coverge for ay a Z ad Z+. If x, y D ψ ad Z +, the x+r, y D ψ for all r R ice x+r + ky x + r + ky, furthermore k x + r ψ + k y Thu the fuctio ψ : D ψ C give by.9 ψx, y + l r mod k ψxy + ky ψx + ly + l ψx + ly

9 ON COVERING EQUIVALENCE 9 i a uiform map ito C. Thi fact wa firt metioed by the author i [S]. ii Let m N. We defie cot m : C C C by m y cot m πx if x Z, m+.0 cot m x, y m y m+ B m+ m+ if x Z & m, 0 if x Z & m, where cot m z dm cot z dz m x C ad y C. A for z C \ πz, ad B i the th Beroulli umber. Fix π cot πv + k v + k v + v v k for v C \ Z, k if x Z the π m+ cot m πx dm dv m + v vx k + d m dv m v + k vx m m! ad o k cot m x, y m! πy m+ + k If x Z ad m, the cot m x, y vaihe ad If x Z ad m, the + k k x + x + k m+ l l 0 cot m x, y m+ πm+ m +! B m+ So we alway have cot m x, y m! πy m+ m! πy m+ + k k x + l l 0 + k d m v dv m v k vx x + k m+ x + k m+. 0. lm+ l m+ m! πy m+ m! x + k m+ π m+ m! m!ζm + πy m+ πy m+ + k k x + k x + k m+. qxy + ky m+.

10 0 ZHI-WEI SUN By part i thi implie that cot m i a uiform map ito C. Let x C ad y > 0. Clearly. Alo, π cot 0 x, y Ix, y π cot πx y Hx, y H x, y ζ x, y ζ x, y.. cot x, y { y cc πx if x Z, 3y if x Z;.3 cot x, y { co πx y 3 i 3 πx if x Z, 0 if x Z, cot 0 x, y cot x, y; ad.4 cot 3 x, y { + co πx y 4 i 4 πx if x Z, 5y 4 if x Z, 6 πy 4 + k k x x + k 4. Remark.3. I 970 S. Chowla [C] proved that if p i a odd prime, the the p real umber cot π r p p r,,, are liearly idepedet over Q, thi wa exteded by T. Okada [O] i 980. By Lemma 7 of Milor [M], there i a uique fuctio g : R R periodic mod for which gx cot m πx m cot m x, for x R \ Z ad.8 hold with θ m+ ad D R, we remark that gx m cot m x, for all x R becaue cot m i a uiform map ito C. With the help of Dirichlet L-fuctio, Milor [M] alo howed that every Q-liear relatio amog the value gx with x Q, follow from.8 with θ m+ ad D Q, ad the fact gx+ gx ad g x m+ gx for x Q. So, each Q-liear relatio amog the value cot m x, m gx with x Q ad m+ Z +, i a coequece of the fact that cot m i a uiform map ito C, together with the trivial equalitie cot m x +, cot m x, ad cot m m cot m. 3. Characterizatio of coverig equivalece I order to characterize coverig equivalece, we eed Lemma 3.. Let f be a complex-valued fuctio o that for ay Z +, fx, i defied ad cotiuou at x, \ Z, ad x + r f, fx, for all x < with x Z.

11 Suppoe that ON COVERING EQUIVALENCE lim fx, for each m N. x m Let A { λ, a, } k be uch a ytem i SC that The A. x + a λ f, 0 for all x < with x Z. Proof. Sice w A i periodic mod [,, k ], it uffice to how w A m 0 for ay m N. A lim x m fx, there exit a δ 0, uch that fx, 0 for all x m δ, m + δ with x m. If Z + the ad hece So Therefore w A m x + {m} lim fx, f x m x + r lim x m λ f r {m} x + {m} f f x+a lim, x m fx, f x+a lim, x m fx,,, fx,, f r {m} m + r a x m. { if m a, 0 otherwie. lim x m fx,, x + a λ f, 0. Let ow characterize the coverig equivalece of two ytem of arithmetic equece. Theorem 3.. Let, Z +, a R ad b t R for,, k ad t,, l. The the followig tatemet are equivalet: 3. A {a } k B {b t } l t; 3. I {,,k} I c I e πi I a J {,,l} t J c J e πi t J b t for all c 0;

12 ZHI-WEI SUN 3.3 k k a z i π S z S z [ z ] l l b t z i π m t t t T z [ z m ] t t T z for z C where S z { k : z a } ad T z { t l : z b t }; 3.4 k U z lt t V z a z b t z m a z b t z t π k l U z [ z ]! [ z ] [ z ] t V z [ z ]! [ z ] m [ z ] t for z C where U z { k : z a + N} ad V z { t l : z b t + N}; 3.5 k u F, v, w + a, l t u v F,, w + b t, for u, v, w C with Rew > Reu + v ad w, w u, w v N. Proof Let N [,, k, m,, m l ]. Set k fz z N/ e πia / ad gz l t z N/ e πib t/. Clearly ay zero of fz or gz i a Nth root of uity. For each a Z, e πia/n i a zero of fz with multiplicity w A a, ad a zero of gz with multiplicity w B a. By Viéte theorem, we have the idetity fz gz if ad oly if w A w B. Note that fz gz if ad oly if I z I N/ e πi I a / I {,,k} J {,,l} J z t J N/ e πi t J b t/. By comparig the coefficiet of power of z, we fid that w A w B if ad oly if I e πi I a / J e πi t J b t/ I {,,k} I N/ a J {,,k} t J N/a for all a 0,,,. Thi prove the equivalece of 3. ad ,3.4. We ca view the multiplicative group C a a Z-module with the calar product m, z z m. By Theorem., for ay z C we have k a z S, l bt z S, t

13 ad ON COVERING EQUIVALENCE 3 k a z γ, l bt z γ,. Apparetly S z T z ad U z V z. If Z +, a R ad z a, the a z z a [ z ]. Therefore 3.3 ad 3.4 follow , ad For Z + ad x, \ Z, we put f x, log i πx, f x, log x + x log logπ. Let j {, }. The lim x m f j x, for all m N. Let Z +. The f j x, i cotiuou for x, \ Z. Whe x < ad x Z, x + r f, log ad x + r f, log If k for x, \ Z, or k x + a i π x + a x+a π t i π x + r log i πx f x, x + r l t l t x+r π log x f x,. π i π x + b t for x, \ Z, the for j or we have x + a f j, l t x + bt m x+b t t π x + bt f j, 0 for all x < with x Z, therefore {, a,,,, a k, k,, b, m,,, b l, m l } by Lemma 3.. So, each of 3.3 ad 3.4 implie Let u, v, w be complex umber with Rew > Reu + v ad w, w u, w v N. By Example.iii, F u,v,w i a uiform map ito the multiplicative group C. Note that 0, D u,v,w. Sice A B, applyig Theorem. we get that k a F u,v,w, l t bt F u.v.w,,

14 4 ZHI-WEI SUN i.e., k F u, v, w + a, l F t u, v, w + b t, Let x R \ Z. By a kow formula i [Ba], if Z + ad a R the F So, by 3.5 we have x, x, + a, k +a x+a x+a +a +a x+a +a x +a x+a. l t x x +a x +b t, x+b t x+b t i.e., k x+a l t x+bt k +a / l t +bt / x+a x+bt. Note that + z zz zz z z for N ad z Z. Let m N. The k + x+m / [m/ ] x+m j0 j k x+a m a + x+m / m a [m/ ]. x+m x+b j0 j t t l m b t Lettig x m we obtai from ad that t l m b t / x + m x + m c for ome c C. k m a t l m b t Thu x+m w Bm w A m ted to a ozero umber a x m. Thi how that w A m w B m. We are doe. Remark 3.. a Whe 3. hold with k < k ad m m l, by takig c / k i 3. we obtai that / k t J / for ome J

15 ON COVERING EQUIVALENCE 5 {,, l} ad hece k m l. Frohi we ca deduce that if A {a } k ad B {b t } l t are equivalet ytem with < < k ad m < < m l the A B i.e. k l, m ad a b for,, k, thi uiquee reult wa dicovered by S. K. Stei [S]. Whe B {b t } l t i the ytem of m copie of 0, the right had ide of the formula i 3. tur out to be { m J if c for ome 0,,, m, J {,,m} J t J c 0 otherwie. Thu. form a exact m-cover of Z if ad oly if I e πi a I m I {,,k} I for,, m ad I {,,k} I J I e πi I a 0 for ay J {,, k} with J Z. For coectio of cover of Z with Egyptia fractio, the reader may ee [Su3 Su7]. b By the proof of ad , 3. i valid if the formula i 3.3 or 3.4 hold for ay z C\Z. Thu 3. ha the followig two equivalet form by aalytic cotiuatio. k l k k z + a i π a z z+a π k l l t i π b t z for all z C; l z + bt t z+b t m m t t for z C \ N. That implie 3. wa firt obtaied by Stei [S] i the cae B {0}; the covere i geeral cae wa oticed by the author i 989 a a coequece of theorem i [Su], whe B {b t } l t i imply {0} it wa foud repeatedly by J. Beebee [Be] i 99. That 3. implie i eetially Corollary 3 of Su [Su] which exted the Gau multiplicatio formula, the covere wa metioed i [Su] a a cojecture. By the above,. i a exact m-cover of Z i.e. A { m, 0, } if ad oly if 3.6 k z + a z+a π k m m z for all z C \ N.

16 6 ZHI-WEI SUN Coequetly, if. i a exact -cover of Z with a 0, the z z+a k z+a π k z/ a k a π k lim z 0 z z z z z for z 0,,,, i.e., 3.7 z z z k z+a z a for all z C \ N by Theorem 3. we directly have k a a / / π k / /. I 994 Beebee [Be3] howed that the relative formula 3.7 hold for ay exact -cover. of Z a we have ee thi wa actually rooted i [Su] publihed i 989. The ew cotributio of [Be3] i that if 3.7 hold the. mut be a exact -cover of Z, however we have a impler equivalet form of 3.. Now we give Theorem 3.. For every,, k we let λ C, Z + ad a R. The the followig tatemet are equivalet: 3.8 A { λ, a, } k ; 3.9 λ t i<j k gcd i, j a i a j λ t i λ t j [ i, j ] for t, where λ Reλ ad λ Imλ for ay,, k; 3.0 [ ] x + ma λ m 0 for all x Z where m i a iteger prime to the moduli,, k ; 3. z + a λ cot m, 0 for all z C where m i a oegative iteger ad cot m z 3. t i a i Example.4ii; z + at λ t ζ, t 0 for all z C \, 0] t

17 ON COVERING EQUIVALENCE 7 where i a complex umber ot i N ad ζ i a i Example.3ii. Proof For t, ad N [,, k ] we ca eaily check that N N λ k r a t λ t + i<j k a i i a j j λ t i λ t j [ i, j ]. So 3.8 ad 3.9 are equivalet Let B { λ, b, } k where b i the leat oegative reidue of ma mod. A m i prime to N [,, k ], ay iteger z ca be writte i the form mu + Nv with u, v Z ad hece w B z w B mu w A u. Thu B if A. Clearly fx, y x ad gx, y fx, y [ ]x, y {x} over R R are uiform map ito R. If A ad x R, the [ ] x + ma λ m x/m + a m λ { } x + b λ 0. If 3.0 hold, the o doe 3.8, becaue for ay x Z we have w A x k ma mx λ [ ma mx λ ] m [ ] ma mx λ [ ] ma mx [ ] ma mx λ m Sice cot m i a uiform map ito C, 3.8 implie 3. by Theorem.. By Example.4ii, cot m z, m! π m+ + k k + z m+ for z C \ Z. Obviouly cot m z, a z ted to a iteger. If Z +, the cot m z, cot m z, / m+ i cotiuou for z C \ Z. I the light of Lemma 3., 3. implie By Example.3ii, ζ i a uiform map ito C. So 3. i implied by 3.8. A 0,,,, by Example.3ii we have d d ζ, v ζ +, v, dv ζ +, v ζ +, v, dv

18 8 ZHI-WEI SUN i the regio C\, 0]. Let m [ Re] if Re, ad m 0 if Re >. Put + m. The Re > ad d m ζ, v dvm 0 j<m If Z + ad a R, the d m z + a dz m ζ, j ζ, v for v C \, 0]. z + a m + ζ, Apparetly ζ z, ζ, z a z ted to a iteger i N. Let aume 3.. The t z + at λ t ζ, t t m + d m dz m t for z C \, 0], ad hece by aalytic cotiuatio t for z C \, 0]. z + at λ t ζ, t 0 t z + at λ t ζ, t 0 for all z 0,,,. t Applyig Lemma 3. we the get 3.8. So far we have completed the proof of Theorem 3.. Remark 3.. I the cae m, that 3.8 implie 3.0 wa firt realized by the author [Su] i 989 ad later refoud by Porubký [P4] i 994. If 3.8 hold, the the formula i 3.0 i the cae m ad x [,, k ] yield the equality k λ 0. I 989 the author [Su] obtaied Theorem. ad oted that fx, y y cot πx over C \ Z C i a uiform map ito C, thu for ay exact -cover. we have cot π z + a cot πz ad for all z C \ Z, thi wa alo give by Beebee [Be] i 99. cc π z + a cc πz Corollary 3.. Let. be a fiite ytem of arithmetic equece, ad m a poitive iteger. The i. form a exact m-cover of Z if ad oly if 3.3 m ad i<j k i, j a i a j [ i, j ] mm,

19 ON COVERING EQUIVALENCE 9 alo. form a exact m-cover of Z if ad oly if 3.4 m ad [ ] a + a am + k m for a RN where i ay fixed iteger prime to N [,, k ]. ii Suppoe that. i a m-cover of Z. The 3.5 t t ζ, x + a t mζ, x for > ad x > 0, t ad for ay Z + we have 3.6 k x+a Z cot π x + a { m cot πx if x R \ Z, 4 B m k x+a if x Z. Proof. Let A {, a,,,, a k, k, m, 0, }. i Clearly A {a } k form a exact m-cover if ad oly if A. If A, the k / m 0 by Remark 3.. That k / m for ay exact m-cover. i actually a well-kow reult, it ca be foud i [P]. By the equivalece of 3.8 ad 3.9, A if ad oly if + m i<j k i, j a i a j [ i, j ] + k, a 0 m Uder the coditio k / m, reduce to the latter equality i 3.3. So, A if ad oly if 3.3 hold. By Theorem 3., A if ad oly if for all x Z we have [ ] x + a m [ ] x + 0 0, i.e. [ ] x + a mx + k m.

20 0 ZHI-WEI SUN Ay iteger x ca be writte i the form a + qn where a R ad q Z, thu the lat equality hold for all x Z if ad oly if 3.4 i valid. Thi ed the proof of part i. ii A w A x m for all x Z, w A x 0 for ay x Z. Obviouly A { w A r, r, N } N. Whe > ad x > 0, by Theorem 3. therefore t x + at ζ, t t t t ζ N x + 0 mζ,, x + a N t mζ, x t ice ζ, x+r N x+r j0 j + N > 0. By Example.4ii, x + r w A rζ N, N, w A r N ζ, x + r 0 N cot x, N! πn + j j x > 0 for all x R. j + x A i the lat paragraph, ow we have x + a cot, m cot x, 0 for ay x R. Clearly thi i equivalet to 3.6. We are doe. Remark i the cae give the followig iequality: 3.7 cc π x + a { m cc πx if x R \ Z, 3 m k if x Z. x+a k x+a Z Let A { λ, a, } k SC ad z C. For y π/ max{,, k }, 0 we have e yz w A e y e y z+a λ e y 0 λ y y z+a ye e y λ y B z+a y! where B x i the Beroulli polyomial of degree. So, A if ad oly if z + λ a B 0 for all 0,,,.

21 ON COVERING EQUIVALENCE For the ytem B {, a,,,, a k, k, m, 0, }, thi wa proved by A. S. Fraekel [F,F] i the cae m ad z 0, by Beebee [Be] i the cae m, ad by Porubký [P] i the cae m Z + ad z 0. See alo Porubký [P,P3] ad Zám [Z] for the cae z 0 with the weight i B replaced by real weight. I 994 Porubký [P4] eetially etablihed the above geeral reult. However, before the work of Beebee [Be] ad Porubký [P4], i 989 the author [Su] proved Theorem. ad oberved that the fuctio b x, y y B x i a uiform map ito C for each N. I 988 D.H. Lehmer [Le] howed that B x i the oly moic polyomial of degree uch that d x + r d B B x for all d Z +. d For ay N clearly Thu A if ad oly if z + λ a B l0 B l l λ l z + a l. 3.9 l0 B l l λ l a l 0 for 0,,,. I 99 E. Y. Deeba ad D. M. Rodriguez [DR] foud thi for the trivial ytem {, 0, d,,, d,,, d, d,, 0, } where d Z +, later Beebee [Be] obtaied the reult for ytem B with m, ad Porubký [P5] oberved the geeralizatio to A SR. For the coverig equivalece betwee ytem i SR, Porubký [P5] provided ome characterizatio ivolvig Euler polyomial ad recurio for Euler umber. I [Su] the author aouced everal reult cloely related to thi paper, proof of them are preeted i a recet paper [Su9]. Referece [B] H. Ba, Geerator ad relatio for cyclotomic uit, Nagoya Math. J , MR 34#98. [Ba] H. Batema, edited by Erdélyi et al., Higher Tracedetal Fuctio, vol. I, McGraw-Hill, 953, Ch. ad. [Be] J. Beebee, Some trigoometric idetitie related to exact cover, Proc. Amer. Math. Soc. 99, MR 9i:03. [Be] J. Beebee, Beroulli umber ad exact coverig ytem, Amer. Math. Mothly 99 99, MR 93i:05. [Be3] J. Beebee, Exact coverig ytem ad the Gau Legedre multiplicatio formula for the gamma fuctio, Proc. Amer. Math. Soc , MR 94f:3300.

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23 ON COVERING EQUIVALENCE 3 [W] H. Walum, Multiplicatio formulae for periodic fuctio, Pacific J. Math , MR 9c:09. [Z] Š.Zám, Vector-coverig ytem of arithmetic equece, Czech. Math. J , MR 50#450. Departmet of Mathematic, Najig Uiverity, Najig 0093, the People Republic of Chia. zwu@ju.edu.c