Abstract
An arithmetic sequence EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$a\left( n \right)=\left\{ a+nx:x\in \mathbb{Z} \right\}\left( 0\le a<n \right)$$ with weight λ ∈ ℂ is denoted by (λ, a, n). For two finite systems EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$A=\left\{ \left\langle {{\lambda }_{s}},{{a}_{s}},{{n}_{s}} \right\rangle \right\}_{s=1}^{k}$$ and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$B=\left\{ \left\langle {{\mu }_{t}},{{b}_{t}},{{m}_{t}} \right\rangle \right\}_{t=1}^{l}$$ of such triples, if EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\sum{_{{{n}_{s}}|x-{{a}_{s}}}{{\lambda }_{s}}}=\sum{_{{{m}_{t}}|x-{{b}_{t}}}}{{\mu }_{t}}$$ for all x ∈ ℤ then we say that A and B are covering equivalent. In this paper we characterize covering equivalence in various ways, our characterizations involve the Γ-function, the Hurwitz ζ-function, trigonometric functions, the greatest integer function and Egyptian fractions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Bass, Generators and relations for cyclotomic units,Nagoya Math. J. 27 (1966), 401–407. MR 34#1298.
H. Bateman, Higher Transcendental Functions (edited by Erdélyi et al), Vol I, McGraw-Hill, 1953, Chapters 1 and 2.
J. Beebee, Some trigonometric identities related to exact covers, Proc. Amer. Math. Soc. 112 (1991), 329–338. MR 91i: 1 1013.
J. Beebee, Bernoulli numbers and exact covering systems, Amer. Math. Monthly 99 (1992), 946–948. MR 93i: 1 1025.
J. Beebee, Exact covering systems and the Gauss-Legendre multiplication formula for the gamma function, Proc. Amer. Math. Soc. 120 (1994), 1061–1065. MR 94f: 33001.
S. D. Chowla, The nonexistence of nontrivial linear relations between the roots of a certain irreducible equation,J. Number Theory 2 (1970), 120–123. MR 40#2638.
E. Y. Deeba and D. M. Rodriguez, Stirling’s series and Bernoulli numbers, Amer. Math. Monthly 98 (1991), 423–426. MR 92g: 1 1025.
V. Ennola, On relations between cyclotomic units,J. Number Theory 4 (1972), 236–247. MR 45#8633 (E 46#1754).
P. Erdös, On integers of the form 2k +p and some related problems, Summa Brasil. Math. 2 (1950), 113–123. MR 13, 437.
A. S. Fraenkel, A characterization of exactly covering congruences,Discrete Math. 4 (1973), 359–366. MR 47#4906.
A. S. Fraenkel, Further characterizations and properties of exactly covering congruences,Discrete Math. 12 (1975), 93–100, 397. MR 51#10276.
R. K. Guy, Unsolved Problems in Number Theory (2nd edition), Springer-Verlag, New York, 1994, Sections A19, B21, E23, F13, F14.
D. S. Kubert, The universal ordinary distribution, Bull. Soc. Math. France 107 (1979), 179–202. MR 81b: 1 2004.
S. Lang, Cyclotomic Fields II (Graduate Texts in Math.; 69), Springer -Verlag, New York, 1980, Ch. 17.
D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly 95 (1988), 905–911. MR 90c: 1 1014.
J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. 29 (1983), 281–322. MR 86d: 1 1007.
T. Okada, On an extension of a theorem of S. Chowla, Acta Arith. 38 (1981), 341–345. MR 83b: 10014.
S. Porubskÿ, Covering systems and generating functions,Acta Arith. 26 (1975), 223–231. MR 52#328.
S. Porubskÿ, On m times covering systems of congruences,Acta Arith. 29 (1976), 159–169. MR 53#2884.
S. Porubskÿ, A characterization of finite unions of arithmetic sequences, Discrete Math. 38 (1982), 73–77. MR 84a: 10057.
S. Porubskÿ, Identities involving covering systems I, Math. Slovaca 44 (1994), 153–162. MR 95f: 1 1002.
S. Porubskÿ, Identities involving covering systems II, Math. Slovaca 44 (1994), 555–568.
Porubskÿ and J. Schönheim, Covering systems of Paul Erdös: past, present and future, preprint, 2000.
S. K. Stein, Unions of arithmetic sequences,Math. Ann. 134 (1958), 289–294. MR 20#17.
Z. W. Sun, Systems of congruences with multipliers, Nanjing Univ. J. Math. Biquarterly 6 (1989), no. 1, 124–133. Zbl. M. 703.11002, MR 90m: 1 1006.
Z. W. Sun, Several results on systems of residue classes, Adv. in Math. (China) 18 (1989), no. 2, 251–252.
Z. W. Sun, On exactly m times covers, Israel J. Math. 77 (1992), 345–348. MR 93k: 1 1007.
Z. W. Sun, Covering the integers by arithmetic sequences, Acta Arith. 72 (1995), 109–129. MR 96k: 1 1013.
Z. W. Sun, Covering the integers by arithmetic sequences II, Trans. Amer. Math. Soc. 348 (1996), 4279–4320. MR 97c: 1 1011.
Z. W. Sun, Exact m-covers and the linear form E x s /n s, Acta Arith. 81 (1997), 175–198. MR 98h: 1 1019.
Z. W. Sun, On covering multiplicity, Proc. Amer. Math. Soc. 127 (1999), 1293–1300. MR 99h: 1 1012.
Z. W. Sun, Products of binomial coefficients modulo p2, Acta Arith. 97 (2001), 87–98.
Z. W. Sun, Algebraic approaches to periodic arithmetical maps, J. Algebra 240 (2001), no. 2, 723–743.
H. Walum, Multiplication formulae for periodic functions, Pacific J. Math. 149 (1991), 383–396. MR 92c: 1 1019.
Š. Znám, Vector-covering systems of arithmetic sequences,Czech. Math. J. 24 (1974), 455–461. MR 50#4520.
Š. Znám, On properties of systems of arithmetic sequences,Acta Arith. 26 (1975), 279–283. MR 51#329.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Sun, ZW. (2002). On Covering Equivalence. In: Jia, C., Matsumoto, K. (eds) Analytic Number Theory. Developments in Mathematics, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3621-2_18
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3621-2_18
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5214-1
Online ISBN: 978-1-4757-3621-2
eBook Packages: Springer Book Archive