episciences.org_7428_1653284221
1653284221
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
05
06
2021
Volume 43  Special...
Partitiontheoretic formulas for arithmetic densities, II
Ken
Ono
Robert
Schneider
Ian
Wagner
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partitiontheoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of qseries 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 partitiontheoretic analogue as well, yielding new qseries density formulas for any subset of N. To do so, we outline a theory of qseries density calculations from first principles, based on a statistic we call the "qdensity" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.
05
06
2021
7428
https://hal.archivesouvertes.fr/hal03208042v1
10.46298/hrj.2021.7428
https://hrj.episciences.org/7428

https://hrj.episciences.org/7428/pdf