A Study on Lambert Series Associated With Cusp Forms and Rankin–Cohen Brackets on Hermitian Jacobi Forms

Abstract

A Lambert series is a series of the form L(q) = ∑∞ n=1 a(n) 1−qn , where a(n) is an arith- metic function and q ∈ C. By setting b(n) = ∑d|n a(d) and q = e−y , the series will take −ny . In 1981, Zagier, conjectured that the Lambert series the form L(y) = ∑∞ n=1 b(n)e 2 −4πny , which is the constant term of the automorphic form y12 |∆(z)|2 , y12 ∑∞ n=1 τ (n)e where ∆(z) is the Ramanujan cusp form of weight 12, has an asymptotic expansion when y → 0+ , and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function ζ (s). In 2000, Hafner and Stopple under the assumption of the Riemann Hypothesis proved this conjecture. In this thesis, we consider a Lambert series asso- ciated to a cusp form and the Möbius function. Using the functional equation of the L-function associated to the cusp form and the functional equation of the Riemann zeta function, we prove an exact formula for the Lambert series. As a consequence, we also derive an asymptotic expansion for the same. We extend our work to higher level cusp forms by considering a more general twisted Lambert series. We also establish an exact formula and asymptotic expansion for a Lambert series associated with the Symmetric square L-function. Rankin–Cohen brackets are bilinear differential operators defined on the space of modular forms. In 2015, Herrero constructed the adjoint map of some linear maps de- fined by using the Rankin–Cohen brackets. In this thesis, we generalize the work of Herrero to the case of Hermitian Jacobi forms over Q(i). Given a fixed Hermitian Ja- cobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin–Cohen brackets. We compute the adjoint maps of such family with respect to the Petersson scalar product. The Fourier coefficients of the Her- mitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.