episciences.org_8932_1653622549
1653622549
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
09
2022
Volume 44  Special...
Power partitions and a generalized eta transformation property
Don
Zagier
In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1q^{n^s}\bigr)^{1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind etafunction in the case $s=1$. This is then combined with the HardyRamanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes
01
09
2022
8932
https://hal.archivesouvertes.fr/hal03494849v1
10.46298/hrj.2022.8932
https://hrj.episciences.org/8932

https://hrj.episciences.org/8932/pdf