Jacobi’s Four and Eight Squares Theorems and Partitions into Distinct Parts

  • Cristina Ballantine
  • Mircea MercaEmail author


We consider the function \(r_s(n)\) which gives the number of ways to write n as the sum of s squares. Since the generating functions for \(r_4(n)\) and \(r_8(n)\) are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolutions involving overpartition functions as well as pentagonal recurrence formulas for \(r_4(n)\) and \(r_8(n)\). These results lead to new connections between divisors and partitions.


Sum of squares Sum of divisors Partitions into distinct parts 

Mathematics Subject Classification

05A17 05A19 11E25 11A25 11P81 



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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceCollege of The Holy CrossWorcesterUSA
  2. 2.Academy of Romanian ScientistsBucharestRomania

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