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into the structure of L2(Γ\G), with Γ = PSL2(Z) and G = PSL2(R). It is shown that there exists a Γ -automorphic function on G, whose value at the unit element is closely related to Z2(g), and whose spectral decomposition in L 2(Γ\G) gives rise to that of Z2(g). This amounts to an alternative and direct proof of the explicit formula for Z2(g) that was… (More)

We study the asymptotics of the lattice point counting function N(x; y; r) = #f° 2 ¡ : d(x; °y)g for X a Riemannian symmetric space of rank one and ¡ a discontinuous group of motions in X, such that ¡nX has ̄nite volume. We show that

- Roelof Bruggeman, T . Mühlenbruch
- 2008

A. The eigenfunctions with eigenvalues 1 or −1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with… (More)

- Roelof Bruggeman, Markus Fraczek, Dieter Mayer
- Experimental Mathematics
- 2013

- Roelof Bruggeman
- 2004

The aim of the present article is to exhibit a new proof of the explicit formula for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in that it dispenses altogether with the spectral theory of sums of Kloosterman sums that played a predominant rôle in the former… (More)

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for SL2 over a totally real number field F , with a discrete subgroup of Hecke type Γ0(I) for a non-zero ideal I in the ring of integers of F . The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many… (More)

In preparation for this review, I decided to remind myself how I became interested in automorphic forms as an undergraduate. I seem to remember that Eric Temple Bell [1] had something to do with it. Imagine my surprise when I found the following statement on page 333 of [1]: “The subject, elliptic functions, in which Jacobi did his first great work, has… (More)

- Nikolaos, Roelof Bruggeman, Nikolaos Diamantis
- 2016

- Roelof Bruggeman
- 1995

We write x = Re z and y = Im z for z ∈ H, and use the Whittaker function W·,·( · ), see, e.g., [12], 1.7. One can express W0,· in terms of a modified Bessel function: W0,μ(y) = √ y/πKμ(y/2). These Maass forms occur as eigenfunctions in the spectral decomposition of the Laplacian in L ( Γmod\H, dxdy y2 ) , with Γmod := PSL2(Z). The eigenvalue is s (1− s).… (More)

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