Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPartition-theoretic Abelian theorems
160
0
0.0
(
0
)
Authors
Robert Schneider
Ask ChatGPT about the research
No Arabic abstract
Using a theorem of Frobenius filtered through partition generating function techniques, we prove partition-theoretic and $q$-series Abelian theorems, yielding analogues of Abels convergence theorem for complex power series, and related formulas. As an application we give a limiting formula for the $q$-bracket of Bloch and Okounkov, an operator from statistical physics connected to the theory of modular forms, as $qto 1$ from within the unit disk.
rate research
Read More
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 $mathbb N$ as limiting values of $q$-series as $qto zeta$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $mathbb N$ by analogous structures in the integer partitions $mathcal P$. In recent work, Wang obtains a wide generalization of Alladis 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 Wangs extension has a partition-theoretic analogue as well, yielding new $q$-series density formulas for any subset of $mathbb N$. 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.
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahons partial fraction decomposition of the generating function for partitions of fixed length.
One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was strengthened by Subbarao in 1966 to say that both $p(2n)$ and $p(2n+1)$ take each value of parity infinitely often. These results have received several other proofs, each relying to some extent on manipulating generating functions. We give a new, self-contained proof of Subbaraos result by constructing a series of bijections and involutions, along the way getting a more general theorem concerning the enumeration of a special subset of integer partitions.
We identify a class of semi-modular forms invariant on special subgroups of $GL_2(mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$ and all column sums in every rectangle equal to a constant $delta in Gamma$. In this paper we prove that for ${a,b} eq{2^{alpha},2k+1}$ where $alpha$ and $k$ are some natural numbers, a $Gamma$-magic rectangle set MRS$_{Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Gamma|$ is odd or $Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Gamma$-magic rectangle MRS$_{Gamma}(a, b)$=MRS$_{Gamma}(a, b;1)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
