### $L$-functions.

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The existence of a strong spectral gap for quotients $\Gamma \setminus G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible...

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature $(n-1,1)$ is “thin”, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg’s theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for ${}_{n}{F}_{n-1}$ are thin.

We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU\left(2\right)$, providing an elementary solution of Ruziewicz problem on ${S}^{2}$ as well as giving many new examples of finitely generated subgroups of $SU\left(2\right)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf{R}[SU(2\left)\right]$ in the $N$-th irreducible representation of $SU\left(2\right)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf{R}[SU(2\left)\right]$, the “unfolded” consecutive spacings...

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