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A computational approach to the generalized Riemann hypothesis

Volume 126 / 2023

Sami Omar, Raouf Ouni Banach Center Publications 126 (2023), 187-204 MSC: Primary 11M06; Secondary 11M26, 11M36. DOI: 10.4064/bc126-11


In this paper, we investigate analytically and numerically the behavior of the Li coefficients $\lambda _{\chi } (n)$ of Dirichlet $L$-functions $L(s, \chi ) $ for large sets of positive integers $n$ and for different primitive characters $\chi $. Actually, we show that the generalized Riemann hypothesis for $L(s,\chi )$ is equivalent to a certain expression of the Li coefficients in terms of a sum of Chebyshev polynomials. The numerical computations based on two different representations of the Li coefficients in terms of a sum over primes or a certain sum of Chebyshev polynomials suggest that the Li coefficients are positive and increasing for $n\geq 1$.


  • Sami OmarDepartment of Mathematics
    University of Bahrain
    P.O. Box 32038 Sukhair, Bahrain
  • Raouf OuniISTLS
    University of Sousse
    4023 Sousse, Tunisia

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