Li coefficients for automorphic L -functions

Jeffrey C. Lagarias[1]

  • [1] University of Michigan Ann Arbor, MI 48109-1043 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 5, page 1689-1740
  • ISSN: 0373-0956

Abstract

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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 ) . Assuming the Riemann hypothesis for L ( s , π ) , we show that λ n ( π ) = N 2 n log n + C 1 ( π ) n + O ( n log n ) , where C 1 ( π ) is a real-valued constant. We construct an entire function F π ( z ) of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for L ( s , π ) , this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in [ - π , π ] having 0 as its only limit point.

How to cite

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Lagarias, Jeffrey C.. "Li coefficients for automorphic $L$-functions." Annales de l’institut Fourier 57.5 (2007): 1689-1740. <http://eudml.org/doc/10275>.

@article{Lagarias2007,
abstract = {Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients $\lambda _n$$(n= 1, 2, \ldots )$. We define similar coefficients $\lambda _n(\pi )$ associated to principal automorphic $L$-functions $L(s, \pi )$ over $GL(N)$. We relate these cofficients to values of Weil’s quadratic functional associated to the representation $\pi $ 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 $L(s, \pi )$. Assuming the Riemann hypothesis for $L(s, \pi )$, we show that $\lambda _n(\pi ) = \frac\{N\}\{2\} n \log n + C_1(\pi ) n + O (\sqrt\{n\}\log \{n\}),$ where $C_1(\pi )$ is a real-valued constant. We construct an entire function $F_\{\pi \}(z)$ of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for $L(s, \pi )$, this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in $[-\pi , \pi ]$ having $0$ as its only limit point.},
affiliation = {University of Michigan Ann Arbor, MI 48109-1043 (USA)},
author = {Lagarias, Jeffrey C.},
journal = {Annales de l’institut Fourier},
keywords = {Automorphic $L$-function; zeta function; automorphic -function},
language = {eng},
number = {5},
pages = {1689-1740},
publisher = {Association des Annales de l’institut Fourier},
title = {Li coefficients for automorphic $L$-functions},
url = {http://eudml.org/doc/10275},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Lagarias, Jeffrey C.
TI - Li coefficients for automorphic $L$-functions
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1689
EP - 1740
AB - Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients $\lambda _n$$(n= 1, 2, \ldots )$. We define similar coefficients $\lambda _n(\pi )$ associated to principal automorphic $L$-functions $L(s, \pi )$ over $GL(N)$. We relate these cofficients to values of Weil’s quadratic functional associated to the representation $\pi $ 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 $L(s, \pi )$. Assuming the Riemann hypothesis for $L(s, \pi )$, we show that $\lambda _n(\pi ) = \frac{N}{2} n \log n + C_1(\pi ) n + O (\sqrt{n}\log {n}),$ where $C_1(\pi )$ is a real-valued constant. We construct an entire function $F_{\pi }(z)$ of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for $L(s, \pi )$, this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in $[-\pi , \pi ]$ having $0$ as its only limit point.
LA - eng
KW - Automorphic $L$-function; zeta function; automorphic -function
UR - http://eudml.org/doc/10275
ER -

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