Hardy Ramanujan Journal |

In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

Source : oai:HAL:hal-03208042v1

Volume: Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan

Published on: May 6, 2021

Submitted on: April 30, 2021

Keywords: integer partitions,arithmetic density,q-series,q-binomial theorem,2010 Mathematics Subject Classification. 05A17, 11P82,[MATH]Mathematics [math]

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