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Register a new userPartition-theoretic formulas for arithmetic densities, II
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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.
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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.
We study M(n,k,r), the number of orbits of {(a_1,...,a_k)in Z_n^k | a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t geq 0} p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at most b parts, each of which is at most a. We derive closed formulas and various identities for such arithmetic partition sums. These results have already appeared in Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and Hermite reciprocity, J. Alg. Combin. 10 (1999) 173-188, and the main result was also published by Von Sterneck in Sitzber. Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601 (see Lemma 2 and references in math.NT/9909121). Thanks to Don Zagier and Robin Chapman for bringing these references to our attention.
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<dots <s_k$ and $ell_1, ell_2,dots , ell_k$. Let $g(mathbf{s}, mathbf{l}, n)$ be the number of $ell$-colored partitions of $n$ with $ell_i$ of the colors appearing only in multiplies of $s_i (1le ile k)$, where $ell = ell_1+cdots +ell_k$. By using the elementary method we obtain an asymptotic formula for the partition function $g(mathbf{s}, mathbf{l}, n)$ with an explicit error term.
Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G_1 under strong assumptions on the ramification in K. Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a different formula for CM(K).G_1 for primitive quartic CM fields with a mild assumption, using a method of proof independent from that of Yang. In this paper we show that these two formulas agree, for a class of primitive quartic CM fields which is slightly larger than the intersection of the fields considered by Yang and Lauter and Viray. Furthermore, the proof that these formulas agree does not rely on the results of Yang or Lauter and Viray. As a consequence of our proof, we conclude that the Bruinier-Yang formula holds for a slightly largely class of quartic CM fields K than what was proved by Yang, since it agrees with the Lauter-Viray formula, which is proved in those cases. The factorization of these intersection numbers has applications to cryptography: precise formulas for them allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography.
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.
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