Partitions with multiplicities associated with divisor functions

Bruce C. Berndt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler

Research output: Contribution to journalArticlepeer-review


We consider colored partitions of a positive integer n, where the number of times a particular colored part m may appear in a partition of n is equal to the sum of the powers of the divisors of m. An asymptotic formula is derived for the number of such partitions. We also derive an asymptotic formula for the number of partitions of n into c colors. In order to achieve the desired bounds on the minor arcs arising from the Hardy-Littlewood circle method, we generalize a bound on an exponential sum twisted by a generalized divisor function due to Motohashi.

Original languageEnglish (US)
Article number127987
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - May 1 2024


  • Colored partitions
  • Estermann–Bettin function
  • Exponential sums
  • Generalized divisor function
  • Hardy-Littlewood circle method
  • Multiplicities

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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