No Arabic abstract
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-type identities to establish families of inequalities for several partition functions. Our proofs heavily rely on some formulas from the work of Zhi-Guo Liu.
In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing invaria
We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields and finitely many weights. We show that for $mathbb Q(sqrt 5)$ there are exactly two such identities.
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $pequiv5pmod8$, then $$prod_{0<x<p/2,(frac{x}{p})=1}xequiv(-1)^{1+r}pmod p,$$ where $(frac{cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelbergers congruence and values of Dirichlet L-series at negative integers.