In a recent important paper, Hoffstein and Hulse [14] generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

An explicit formula is given for a sequence of numbers by using the Eichler-Selberg trace formula. A criterion is obtained by using these numbers for the location of nontrivial zeros of Hecke L-functions, which are associated with a family of cusp forms of weight 2.

Let X = Γ\G/K be a compact locally symmetric space. In this paper we establish a version of the Selberg trace formula for non-unitary representations of the lattice Γ. On the spectral side appears the spectrum of the “flat Laplacian” ∆, acting in the space of sections of the associated flat bundle. In general, this is a non-self-adjoint operator.

The thermodynamics of interacting vortices constricted to move in one-dimensional tracks carved out on superconducting films in the extreme type-II limit is mapped into the Coulomb gas model of random matrices. Application of the Selberg Integral supply exact expressions for the constitutive relation for two configurations of tracks. PACS number: 74.20.Hi, 02.10.Sp To appear in J. Phys. Cond. M...

This paper is an expository account of our 1976 monograph [6] on Scattering theory for automorphic functions. Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an explicit formula for the translation representations and a simpler derivation of the spectral representations. Our hyperbolic approach to the Selberg trace formula ...

We specify sufficient conditions for the square modulus of the local parameters of a family of GLn cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≤ 4. As an application, we show that Rankin-Selberg L-functions on GLn1 × GLn2 , for ni ≤ 4, satisfy the standard convexity bound.

The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the two-point spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos applications and can serve as a non-formal introduction to mathematical methods of quantum chaos.

We consider the family of Rankin-Selberg convolution L-functions of a fixed SL(3,Z) Maass form with the family of Hecke-Maass cusp forms on SL(2,Z). We estimate the second moment of this family of L-functions with a “long” integration in t-aspect. These L-functions are distinguished by their high degree (12) and large conductors (of size T ).