A relation between Fourier coefficients of holomorphic cusp forms and exponential sums

Anne-Maria Ernvall-Hytönen

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Let $R (λ)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that $\int_{1}^{T} {| R (t) |}^{2} d t = c T^{\frac{5}{2}} + O_{δ} (T^{\frac{9}{4} + δ})$ , where $c$ is a specific nonzero constant and $δ$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R (t) = O_{δ} (t^{\frac{3}{4} + δ})$ .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2 n + 1$ -dimensional case. ...