The first eigenvalue problem and tensor products of zeta functions
The paper under review is concerned with Selberg's eigenvalue conjecture for the lowest positive eigenvalue \(\lambda_1\) of the Laplacian on a congruence subgroup of \(\text{SL}_2(\mathbb{Z})\) or \(\text{SL}_2({\mathcal O}_\alpha)\), where \({\mathcal O}_\alpha\) is the ring of integers of an imaginary quadratic number field of discriminant \(d<0\). The author introduces the following assumption on the integer \(n\geq 2\) and a cuspidal automorphic representation \(\pi\) of \(\text{GL}_2(\mathbb{A}_k)\) with a number field \(k\): There is a cuspidal automorphic representation \(\pi_n=\text{Sym}^{n-1}(\pi)\) of \(\text{GL}_n(\mathbb{A}_k)\) whose \(L\)-function is equal to \(L(s,\text{Sym}^{n-1}(\pi))\). This assumption is known to hold for \(n\geq 5\). If this assumption is true for all \(n\) then Selberg's conjecture holds true. Moreover, if this assumption holds for some integer \(n\geq 2\) the author gives lower bounds for \(\lambda_1\) depending on \(n\) which give the best known lower bounds in the two-dimensional case (for \(n=4\) or 5, respectively). For \(n=5\) this bound yields in the three-dimensional case (i.e. for congruence subgroups of \(\text{SL}_2({\mathcal O}_d)): \lambda_1\geq \frac{160}{169}=0.946\dots\) which is remarkably close to Selberg's conjecture \(\lambda_1\geq 1\). In the last section the author comments on the general principle that tensor products of zeta-functions will be useful for solving problems of the type of the Riemann hypothesis, and he introduces a possibly useful candidate.
scientific article; zbMATH DE number 2138332