The Lambert series factorization theorem
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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.
KeywordsDivisors Lambert series Partitions
Mathematics Subject Classification11P81 11A25 11P84 05A17 05A19
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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