Displaying similar documents to “On Li’s Coefficients for Some Classes of L-Functions”

Higher regularizations of zeros of cuspidal automorphic L -functions of GL d

Masato Wakayama, Yoshinori Yamasaki (2011)

Journal de Théorie des Nombres de Bordeaux

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We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic L -functions of GL d over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.

Li coefficients for automorphic L -functions

Jeffrey C. Lagarias (2007)

Annales de l’institut Fourier

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Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients λ n ( n = 1 , 2 , ... ) . We define similar coefficients λ n ( π ) associated to principal automorphic L -functions L ( s , π ) over G L ( N ) . We relate these cofficients to values of Weil’s quadratic functional associated to the representation π on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for L ( s , π ) ....

Artin formalism for Selberg zeta functions of co-finite Kleinian groups

Eliot Brenner, Florin Spinu (2009)

Journal de Théorie des Nombres de Bordeaux

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Let Γ ∖ ℍ 3 be a finite-volume quotient of the upper-half space, where Γ ⊂ SL ( 2 , ℂ ) is a discrete subgroup. To a finite dimensional unitary representation χ of Γ one associates the Selberg zeta function Z ( s ; Γ ; χ ) . In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if Γ ˜ is a finite index group extension of Γ in SL ( 2 , ℂ ) , and π = Ind Γ Γ ˜ χ is the induced representation, then Z ( s ; Γ ; χ ) = Z ( s ; Γ ˜ ; π ) . In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely φ ( s ; Γ ; χ ) = φ ( s ; Γ ˜ ; π ) ,...