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On Li’s coefficients for the Rankin-Selberg \(L\)-functions. (English) Zbl 1248.11036
In 1997, X.-J. Li [J. Number Theory 65, No. 2, 325–333 (1997; Zbl 0884.11036)] studied the sequence \[ \lambda_n = \lim_{T\to \infty} \sum_{|\text{Im}\,\rho |\leq T}\left(1-\left(1-\frac1\rho\right)^n\right), \] where \(\rho\) runs over non-trivial zeros of the Riemann zeta-function. (The first step is to show that \(\lambda_n\) is well-defined, i.e., that the limit indeed converges, for all \(n\).) Li showed that the non-negativity of \(\lambda_n\) for all \(n\) is equivalent to the Riemann hypothesis. The numbers \(\lambda_n, \;n=1, 2, \dots\) have since become known as Li coefficients. By replacing the Riemann zeta-function with some other automorphic \(L\) function, one may define generalized Li coefficients. The present paper is concerned with generalized Li coefficients attached to the Rankin-Selberg convolution \(L\) function \(L(s, \pi \times \pi'),\) where \(\pi\) and \(\pi'\) are irreducible (unitary) cuspidal automorphic representations of \(\text{GL}_m(\mathbb A_F)\) and \(\text{GL}_{m'}(\mathbb A_F)\) and \(\mathbb A_F\) is the adele ring of a number field \(F.\)
The authors first give an estimate for the number of nontrivial zeros of \(L(s, \pi \times \pi')\) up to height \(T\) and a version of the Weil explicit formula for \(L(s, \pi \times \pi').\) This is then used to prove the well-definedness of the the generalized Li coefficients (i.e., convergence of the above limit in this case). The next step is to give an arithmetic expression for the generalized Li coefficients, extending the work of E. Bombieri and J. C. Lagarias [J. Number Theory 77, No. 2, 274–287 (1999; Zbl 0972.11079)] for the original Li coefficients, and J. Lagarias [Ann. Inst. Fourier 57, No. 5, 1689–1740 (2007; Zbl 1216.11078)] for generalized Li coefficients attached to \(\text{GL}_n\) standard \(L\) functions. This is done in two different ways, and includes a decomposition of the generalized Li coefficients into finite and archimedean contributions. Then, an asymptotic expansion for the archimedean contribution as well as an expression for the finite part is obtained. This is an extension and generalization of further work of Lagarias [ibid].
Finally, the authors evaluate the Li coefficients by a second method, which was suggested by J. Oesterlé (see A. Voros [Math. Phys. Anal. Geom. 9, No. 1, 53–63 (2006; Zbl 1181.11055)]) and requires the generalized Riemann hypothesis, and obtain a bound towards the generalized Ramanujan conjecture in the archimedean component.

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M41 Other Dirichlet series and zeta functions
Full Text: DOI
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