Abstract
In this paper, we prove under the Riemann hypothesis that the Li coefficients for the Dirichlet \(L\)-functions \(\lambda _{\chi }(n)\) are increasing in \(n\). We also prove unconditionally that the first Li coefficients are increasing using the Bell polynomials. Furthermore, we give a probabilistic interpretation and describe another method differently as stated in Omar et al. (LMS J Comput Math 14:140–154, 2011) to compute them.
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Acknowledgments
I am grateful to Professor Emmanuel Royer who has pointed to me the recent work of Reyna [16]. We also thank Professor Sami Omar for posing this problem and for many helpful discussions and the referee for many valuable suggestions that increased the clarity of the presentation.
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Communicated by A. Constantin.
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Mazhouda, K. On the Li coefficients for the Dirichlet \(L\)-functions. Monatsh Math 170, 405–423 (2013). https://doi.org/10.1007/s00605-012-0436-3
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DOI: https://doi.org/10.1007/s00605-012-0436-3