[C] A. Selberg, Harmonic analysis and discon-tinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet series ,

We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto’s factorization results for biregular bipartite graphs apply, leading to exact factorizations. For (d, r)-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Solé. Finally, we give an example to show how the...

Duke and Kowalski in [6] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin-Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [6]. As an application, we give a generalization to GL(n) of Duke’s multiplicity theor...

A new type of sl3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl3 basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the sl3 Selberg integrals of Tarasov and Varchenko, and to alternati...

[C] A. Selberg, Harmonic analysis and discon-tinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet series ,

[C] A. Selberg, Harmonic analysis and discon-tinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet series ,

In this paper, new examples of weakly symmetric spaces in the sense of A. Selberg 14] are constructed. Let G be a connected, simply-connected, simple Lie group of hermitian type and K a maximal compactly embedded subgroup of G such that D = G=K is an irreducible classical bounded symmetric domain. Let G 1 and D 1 be circle extensions of G and D, respectively. The factor of automorphy induces a ...

with fκ(s) as large as possible (resp. Fκ(s) as small as possible) given that the above inequality holds for all choices of A satisfying (1). Selberg [3] has shown (in a much more general context) that the functions fκ(s), Fκ(s) are continuous, monotone, and computable for s > 1, and that they tend to 1 exponentially as s goes to infinity. Let β = β(κ) be the infimum of s such that fκ(s) > 0. S...

We prove the Rankin-Selberg convolution of two cuspidal automorphic representations are automorphic whenever one of them arises from an irreducible representation of an abelian-by-nilpotent Galois extension, which extends the previous result of Arthur-Clozel. Moreover, if one of such representations is of dimension at most 3 and another representation arises from a nearly nilpotent extension or...

We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur’s identity on the characteristic polynomial of a sum of matrices. The second evaluation of the ...