New Connections Between Functions from Additive and Multiplicative Number Theory
In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and q -series methods. These general results are illustrated con...
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| Veröffentlicht in: | Mediterranean journal of mathematics Jg. 15; H. 2; S. 1 - 13 |
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| Abstract | In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and
q
-series methods. These general results are illustrated considering relationships between the gcd-sum function and integer partitions. Even if there is in the literature a large number of works in which many properties of the gcd-sum function are studied, connections between the gcd-sum function and integer partitions have not been remarked so far. Our general results can be used to provide new connections between the partitions and many classical special arithmetic functions often studied in multiplicative number theory: the Möbius function
μ
(
n
)
, Euler’s totient
φ
(
n
)
, Jordan’s totient
J
k
(
n
)
, Liouville’s function
λ
(
n
)
, the von Mangoldt function
Λ
(
n
)
, the divisor function
σ
x
(
n
)
, and others. |
|---|---|
| AbstractList | In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and q-series methods. These general results are illustrated considering relationships between the gcd-sum function and integer partitions. Even if there is in the literature a large number of works in which many properties of the gcd-sum function are studied, connections between the gcd-sum function and integer partitions have not been remarked so far. Our general results can be used to provide new connections between the partitions and many classical special arithmetic functions often studied in multiplicative number theory: the Möbius function μ(n), Euler’s totient φ(n), Jordan’s totient Jk(n), Liouville’s function λ(n), the von Mangoldt function Λ(n), the divisor function σx(n), and others. In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and q -series methods. These general results are illustrated considering relationships between the gcd-sum function and integer partitions. Even if there is in the literature a large number of works in which many properties of the gcd-sum function are studied, connections between the gcd-sum function and integer partitions have not been remarked so far. Our general results can be used to provide new connections between the partitions and many classical special arithmetic functions often studied in multiplicative number theory: the Möbius function μ ( n ) , Euler’s totient φ ( n ) , Jordan’s totient J k ( n ) , Liouville’s function λ ( n ) , the von Mangoldt function Λ ( n ) , the divisor function σ x ( n ) , and others. |
| ArticleNumber | 36 |
| Author | Merca, Mircea |
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| References | KiuchiISums of averages of gcd-sum functionsJ. Number Theory2017176449472362213910.1016/j.jnt.2016.12.02106697572 Tóth, L.: On the bi-unitary analogues of Eulers arithmetical function and the gcd-sum function. J. Integer Seq. 12, Article 09.5.2 (2009) CohenEArithmetical functions of a greatest common divisor, IProc. Am. Math. Soc.1960111641711117130096.02906 TóthLThe unitary analogue of Pillai’s arithmetical functionCollect. Math.198940193010780890712.11010 SivaramakrishnanROn three extensions of Pillai’s arithmetic function β(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (n)$$\end{document}Math. Stud.1971391871903377390261.10006 Merca, M., Schmidt, M.D.: A partition identity related to Stanley’s theorem. Am. Math. Mon (accepted, to appear) Merca, M., Schmidt, M.D.: The partition function p(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(n)$$\end{document} in terms of the classical Möbius function. Ramanujan J. (accepted, to appear) SchulteJÜber die Jordansche Verallgemeinerung der Eulerschen FunktionResults Math.199936354364172621410.1007/BF033221220942.11006 Zhang, D., Zhai, W.: Mean values of a gcd-sum function over regular integers modulo n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}. J. Integer Seq. 13, Article 10.4.7 (2010) PillaiSSOn an arithmetic functionJ. Annamalai Univ.193322432480008.19603 Bordellès, O.: Mean values of generalized gcd-sum and lcm-sum functions. J. Integer Seq. 10, Article 07.9.2 (2007) CesàroEÉtude moyenne du plus grand commun diviseur de deux nombresAnn. Mat. Pura Appl. (1)18851323525010.1007/BF0242080017.0144.05 Tanigawa, Y., Zhai, W.: On the gcd-sum function. J. Integer Seq. 11, Article 08.2.3 (2008) Tóth, L.: A survey of gcd-sum functions. J. Integer Seq. 13, Article 10.8.1 (2010) CohenEArithmetical functions of a greatest common divisor, II. An alternative approachBoll. Un. Mat. Ital.1962173493561500930109.27402 SándorJKrámerA-VÜber eine zahlentheoretische FunktionMath. Morav.1999353620972.11005 TóthLThe unitary analogue of Pillai’s arithmetical function IINotes Number Theory Discrete Math.1996240461418833 Bordellès, O.: A note on the average order of the gcd-sum function. J. Integer Seq. 10, Article 07.3.3 (2007) Tóth, L.: Weighted gcd-sum functions. J. Integer Seq. 14, 11.7.7 (2011) Bordellès, O.: The composition of the gcd and certain arithmetic functions. J. Integer Seq. 13, Article 10.7.1 (2010) Broughan, K.A.: The gcd-sum function. J. Integer Seq. 4, Article 01.2.2 (2001) AndrewsGEThe Theory of Partitions1976New YorkAddison-Wesley Publishing0371.10001 Tóth, L.: A gcd-sum function over regular integers (mod n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}). J. Integer Seq. 12, Article 09.2.5 (2009) Broughan, K.A.: The average order of the Dirichlet series of the gcd-sum function. J. Integer Seq. 10, Article 07.4.2 (2007) MercaMThe Lambert series factorization theoremRamanujan J.2017442417435371542410.1007/s11139-016-9856-306832115 ChidambaraswamyJSitaramachandraraoRAsymptotic results for a class of arithmetical functionsMonatsh. Math.198599192777816710.1007/BF013007350551.10035 DicksonLEHistory of the Theory of Numbers1971New YorkChelsea HaukkanenPRational arithmetical functions of order (2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,1)$$\end{document} with respect to regular convolutionsPortugal. Math.19995632934317160570939.11003 HaukkanenPOn a gcd-sum functionAequationes Math.200876168178244346810.1007/s00010-007-2923-51243.11004 Cohen, E.: Arithmetical functions of a greatest common divisor, III. Cesàro’s divisor problem. Proc. Glasgow Math. Assoc. 5, 67–75 (1961–1962) J Sándor (1091_CR20) 1999; 3 SS Pillai (1091_CR19) 1933; 2 J Chidambaraswamy (1091_CR8) 1985; 99 1091_CR17 1091_CR18 M Merca (1091_CR16) 2017; 44 1091_CR30 1091_CR10 P Haukkanen (1091_CR14) 2008; 76 P Haukkanen (1091_CR13) 1999; 56 L Tóth (1091_CR24) 1989; 40 E Cesàro (1091_CR7) 1885; 13 I Kiuchi (1091_CR15) 2017; 176 J Schulte (1091_CR22) 1999; 36 LE Dickson (1091_CR12) 1971 R Sivaramakrishnan (1091_CR21) 1971; 39 L Tóth (1091_CR25) 1996; 2 1091_CR27 1091_CR28 1091_CR29 E Cohen (1091_CR11) 1962; 17 1091_CR23 1091_CR26 1091_CR6 GE Andrews (1091_CR1) 1976 1091_CR5 1091_CR4 1091_CR3 1091_CR2 E Cohen (1091_CR9) 1960; 11 |
| References_xml | – reference: Bordellès, O.: Mean values of generalized gcd-sum and lcm-sum functions. J. Integer Seq. 10, Article 07.9.2 (2007) – reference: Bordellès, O.: The composition of the gcd and certain arithmetic functions. J. Integer Seq. 13, Article 10.7.1 (2010) – reference: Cohen, E.: Arithmetical functions of a greatest common divisor, III. Cesàro’s divisor problem. Proc. Glasgow Math. Assoc. 5, 67–75 (1961–1962) – reference: Tanigawa, Y., Zhai, W.: On the gcd-sum function. J. Integer Seq. 11, Article 08.2.3 (2008) – reference: Tóth, L.: A survey of gcd-sum functions. J. Integer Seq. 13, Article 10.8.1 (2010) – reference: MercaMThe Lambert series factorization theoremRamanujan J.2017442417435371542410.1007/s11139-016-9856-306832115 – reference: DicksonLEHistory of the Theory of Numbers1971New YorkChelsea – reference: KiuchiISums of averages of gcd-sum functionsJ. Number Theory2017176449472362213910.1016/j.jnt.2016.12.02106697572 – reference: CohenEArithmetical functions of a greatest common divisor, IProc. Am. Math. Soc.1960111641711117130096.02906 – reference: HaukkanenPRational arithmetical functions of order (2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,1)$$\end{document} with respect to regular convolutionsPortugal. Math.19995632934317160570939.11003 – reference: SchulteJÜber die Jordansche Verallgemeinerung der Eulerschen FunktionResults Math.199936354364172621410.1007/BF033221220942.11006 – reference: HaukkanenPOn a gcd-sum functionAequationes Math.200876168178244346810.1007/s00010-007-2923-51243.11004 – reference: TóthLThe unitary analogue of Pillai’s arithmetical functionCollect. Math.198940193010780890712.11010 – reference: PillaiSSOn an arithmetic functionJ. Annamalai Univ.193322432480008.19603 – reference: CohenEArithmetical functions of a greatest common divisor, II. An alternative approachBoll. Un. Mat. Ital.1962173493561500930109.27402 – reference: AndrewsGEThe Theory of Partitions1976New YorkAddison-Wesley Publishing0371.10001 – reference: CesàroEÉtude moyenne du plus grand commun diviseur de deux nombresAnn. Mat. Pura Appl. (1)18851323525010.1007/BF0242080017.0144.05 – reference: TóthLThe unitary analogue of Pillai’s arithmetical function IINotes Number Theory Discrete Math.1996240461418833 – reference: SivaramakrishnanROn three extensions of Pillai’s arithmetic function β(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (n)$$\end{document}Math. Stud.1971391871903377390261.10006 – reference: Broughan, K.A.: The gcd-sum function. J. Integer Seq. 4, Article 01.2.2 (2001) – reference: Merca, M., Schmidt, M.D.: The partition function p(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(n)$$\end{document} in terms of the classical Möbius function. Ramanujan J. (accepted, to appear) – reference: Tóth, L.: On the bi-unitary analogues of Eulers arithmetical function and the gcd-sum function. J. Integer Seq. 12, Article 09.5.2 (2009) – reference: Zhang, D., Zhai, W.: Mean values of a gcd-sum function over regular integers modulo n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}. J. Integer Seq. 13, Article 10.4.7 (2010) – reference: ChidambaraswamyJSitaramachandraraoRAsymptotic results for a class of arithmetical functionsMonatsh. Math.198599192777816710.1007/BF013007350551.10035 – reference: Bordellès, O.: A note on the average order of the gcd-sum function. J. Integer Seq. 10, Article 07.3.3 (2007) – reference: Broughan, K.A.: The average order of the Dirichlet series of the gcd-sum function. J. Integer Seq. 10, Article 07.4.2 (2007) – reference: SándorJKrámerA-VÜber eine zahlentheoretische FunktionMath. Morav.1999353620972.11005 – reference: Tóth, L.: A gcd-sum function over regular integers (mod n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}). J. Integer Seq. 12, Article 09.2.5 (2009) – reference: Tóth, L.: Weighted gcd-sum functions. J. Integer Seq. 14, 11.7.7 (2011) – reference: Merca, M., Schmidt, M.D.: A partition identity related to Stanley’s theorem. Am. Math. Mon (accepted, to appear) – volume: 99 start-page: 19 year: 1985 ident: 1091_CR8 publication-title: Monatsh. Math. doi: 10.1007/BF01300735 – volume: 40 start-page: 19 year: 1989 ident: 1091_CR24 publication-title: Collect. Math. – ident: 1091_CR5 – volume: 44 start-page: 417 issue: 2 year: 2017 ident: 1091_CR16 publication-title: Ramanujan J. doi: 10.1007/s11139-016-9856-3 – ident: 1091_CR29 – volume-title: The Theory of Partitions year: 1976 ident: 1091_CR1 – ident: 1091_CR10 doi: 10.1017/S2040618500034328 – ident: 1091_CR27 – ident: 1091_CR23 – volume: 39 start-page: 187 year: 1971 ident: 1091_CR21 publication-title: Math. Stud. – volume: 56 start-page: 329 year: 1999 ident: 1091_CR13 publication-title: Portugal. Math. – volume: 76 start-page: 168 year: 2008 ident: 1091_CR14 publication-title: Aequationes Math. doi: 10.1007/s00010-007-2923-5 – volume: 3 start-page: 53 year: 1999 ident: 1091_CR20 publication-title: Math. Morav. – ident: 1091_CR3 – ident: 1091_CR17 – ident: 1091_CR30 – volume: 13 start-page: 235 year: 1885 ident: 1091_CR7 publication-title: Ann. Mat. Pura Appl. (1) doi: 10.1007/BF02420800 – ident: 1091_CR6 – volume: 176 start-page: 449 year: 2017 ident: 1091_CR15 publication-title: J. Number Theory doi: 10.1016/j.jnt.2016.12.021 – volume: 11 start-page: 164 year: 1960 ident: 1091_CR9 publication-title: Proc. Am. Math. Soc. – ident: 1091_CR26 – volume: 17 start-page: 349 year: 1962 ident: 1091_CR11 publication-title: Boll. Un. Mat. Ital. – ident: 1091_CR28 – volume-title: History of the Theory of Numbers year: 1971 ident: 1091_CR12 – ident: 1091_CR18 – volume: 36 start-page: 354 year: 1999 ident: 1091_CR22 publication-title: Results Math. doi: 10.1007/BF03322122 – volume: 2 start-page: 40 year: 1996 ident: 1091_CR25 publication-title: Notes Number Theory Discrete Math. – volume: 2 start-page: 243 year: 1933 ident: 1091_CR19 publication-title: J. Annamalai Univ. – ident: 1091_CR2 – ident: 1091_CR4 |
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