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The Ramanujan Journal

, Volume 44, Issue 2, pp 417–435 | Cite as

The Lambert series factorization theorem

  • Mircea Merca
Article

Abstract

A factorization for partial sums of Lambert series is introduced in this paper. As corollaries, we derive some connections between partitions and divisors. These results can be easily used to discover and prove new combinatorial identities involving important functions from 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 \(\Lambda (n)\), and the divisor function \(\sigma _x(n)\). The fascinating feature of these identities is their common nature.

Keywords

Divisors Lambert series Partitions 

Mathematics Subject Classification

11P81 11A25 11P84 05A17 05A19 

Notes

Acknowledgements

The author likes to thank the referees for their helpful comments. The author also likes to mention his special thanks to Dr. Oana Merca for the careful reading of the manuscript and helpful remarks.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

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