An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

@article{Li2004AnAF,
  title={An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials},
  author={Xian-jin Li},
  journal={Journal of Number Theory},
  year={2004},
  volume={113},
  pages={175-200}
}
  • Xian-jin Li
  • Published 9 March 2004
  • Mathematics
  • Journal of Number Theory
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References

SHOWING 1-10 OF 22 REFERENCES
Explicit formulas for Dirichlet and Hecke $L$-functions
In 1997, the author proved that the Riemann hypothesis holds if and only if λn = ∑ [1−(1−1/ρ)n] > 0 for all positive integers n, where the sum is over all complex zeros of the Riemann zeta function.
Complements to Li's Criterion for the Riemann Hypothesis☆
Abstract In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only ifλn=∑ρ [1−(1−1/ρ)n] hasλn>0 forn=1, 2, 3, … whereρruns over the complex zeros of the Riemann zeta
An explicit formula for Hecke $L$-functions
In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials,
Introduction to the arithmetic theory of automorphic functions
* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary
Topics in classical automorphic forms
Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators
Automorphic forms on Adele groups
This volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of
The Positivity of a Sequence of Numbers and the Riemann Hypothesis
Abstract In this note, we prove that the Riemann hypothesis for the Dedekind zeta function is equivalent to the nonnegativity of a sequence of real numbers.
La conjecture de Weil. I
© Publications mathematiques de l’I.H.E.S., 1974, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www.
MATH
TLDR
It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.
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