Narrow Results

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.

We show that one simple zero of a Maass L-function implies infinitely many simple zeros of the L-function. Since Strömbergsson found simple zeros of three different even Maass L-functions, we know that there are at least three Maass L-functions having infinitely many simple zeros.

We obtain certain estimates for averages of shifted convolution sums involving the Fourier coefficients of a normalized Hecke–Maass eigenform and holomorphic cusp form.

Let $\mathcal{𝒜}(n)$ be the $(1,n)$th Fourier coefficient of $SL(3,\mathbb{Z})$ Hecke–Maass cusp form, denoted as $A(1,n)$ or the triple divisor function, denoted as $d_3(n)$. Let $k\ge 3$ be an integer. In this paper, we establish an asymptotic formula for the sum