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 \(\mu (n)\), Euler’s totient \(\varphi (n)\), Jordan’s totient \(J_k(n)\), Liouville’s function \(\lambda (n)\), the von Mangoldt function \(\Lambda (n)\), the divisor function \(\sigma _x(n)\), and others.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing, New York (1976)
Bordellès, O.: A note on the average order of the gcd-sum function. J. Integer Seq. 10, Article 07.3.3 (2007)
Bordellès, O.: Mean values of generalized gcd-sum and lcm-sum functions. J. Integer Seq. 10, Article 07.9.2 (2007)
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)
Broughan, K.A.: The average order of the Dirichlet series of the gcd-sum function. J. Integer Seq. 10, Article 07.4.2 (2007)
Cesàro, E.: Étude moyenne du plus grand commun diviseur de deux nombres. Ann. Mat. Pura Appl. (1) 13, 235–250 (1885)
Chidambaraswamy, J., Sitaramachandrarao, R.: Asymptotic results for a class of arithmetical functions. Monatsh. Math. 99, 19–27 (1985)
Cohen, E.: Arithmetical functions of a greatest common divisor, I. Proc. Am. Math. Soc. 11, 164–171 (1960)
Cohen, E.: Arithmetical functions of a greatest common divisor, III. Cesàro’s divisor problem. Proc. Glasgow Math. Assoc. 5, 67–75 (1961–1962)
Cohen, E.: Arithmetical functions of a greatest common divisor, II. An alternative approach. Boll. Un. Mat. Ital. 17, 349–356 (1962)
Dickson, L.E.: History of the Theory of Numbers, vol. I. Chelsea, New York (1971)
Haukkanen, P.: Rational arithmetical functions of order \((2,1)\) with respect to regular convolutions. Portugal. Math. 56, 329–343 (1999)
Haukkanen, P.: On a gcd-sum function. Aequationes Math. 76, 168–178 (2008)
Kiuchi, I.: Sums of averages of gcd-sum functions. J. Number Theory 176, 449–472 (2017)
Merca, M.: The Lambert series factorization theorem. Ramanujan J. 44(2), 417–435 (2017)
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 \(p(n)\) in terms of the classical Möbius function. Ramanujan J. (accepted, to appear)
Pillai, S.S.: On an arithmetic function. J. Annamalai Univ. 2, 243–248 (1933)
Sándor, J., Krámer, A.-V.: Über eine zahlentheoretische Funktion. Math. Morav. 3, 53–62 (1999)
Sivaramakrishnan, R.: On three extensions of Pillai’s arithmetic function \(\beta (n)\). Math. Stud. 39, 187–190 (1971)
Schulte, J.: Über die Jordansche Verallgemeinerung der Eulerschen Funktion. Results Math. 36, 354–364 (1999)
Tanigawa, Y., Zhai, W.: On the gcd-sum function. J. Integer Seq. 11, Article 08.2.3 (2008)
Tóth, L.: The unitary analogue of Pillai’s arithmetical function. Collect. Math. 40, 19–30 (1989)
Tóth, L.: The unitary analogue of Pillai’s arithmetical function II. Notes Number Theory Discrete Math. 2, 40–46 (1996)
Tóth, L.: A gcd-sum function over regular integers (mod \(n\)). J. Integer Seq. 12, Article 09.2.5 (2009)
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)
Tóth, L.: A survey of gcd-sum functions. J. Integer Seq. 13, Article 10.8.1 (2010)
Tóth, L.: Weighted gcd-sum functions. J. Integer Seq. 14, 11.7.7 (2011)
Zhang, D., Zhai, W.: Mean values of a gcd-sum function over regular integers modulo \(n\). J. Integer Seq. 13, Article 10.4.7 (2010)
Acknowledgements
The author would like to thank the referees for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Merca, M. New Connections Between Functions from Additive and Multiplicative Number Theory. Mediterr. J. Math. 15, 36 (2018). https://doi.org/10.1007/s00009-018-1091-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-018-1091-2