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Register a new userAsymptotic formulas for general colored partition functions
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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.
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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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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.
We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujans mock theta functions $f(q)$ and $omega(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products.
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