Related papers: Overpartitions and class numbers of binary quadrat…

In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of…

This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…

Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of…

In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of…

We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic…

We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their…

In this paper, we use theta integrals to give a different construction of mock Maass forms studied by Sander Zwegers. With this method, we construct new real-analytic modular forms, whose Fourier coefficients are logarithms of algebraic…

Shape theory works nice for (Hausdorff) paracompact spaces, but for spaces with no separation axioms, it seems to be quite poor. However, for finite and locally finite spaces their weak homotopy type is rather rich, and is equivalent to the…

The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two cut out by certain linear relations between their Fourier coefficients. We define an analogous quaternionic Maass…

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the…

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on…

This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of…

The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms which is an extension of the Borcherd-Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this…

We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $\mathbb{Q}(\sqrt{-7})$ and $\mathbb{Q}(\sqrt{-11})$. In both cases we prove that the subrings of symmetric modular…

We show that the meromorphic Jacobi form that counts the quarter-BPS states in N=4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes…

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the…

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of…

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson,…

In this paper we give some evidence for the Tate (and Hodge) conjecture(s) for a class of Hilbert modular fourfolds X, whose connected components arise as arithmetic quotients of the fourfold product of the upper half plane by congruence…

We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers-Ramanujan partitions, theta functions, mock theta functions, partitions with…