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**On asymptotic behavior of generalized Li coefficients.**
*(English)*
Zbl 1429.11172

Summary: In this paper, we consider the asymptotic behaviour of \(\tau\)-Li coefficients for the wide class of \(L\)-functions that contains the Selberg class, the class of all automorphic \(L\)-functions, the Rankin-Selberg \(L\)-functions, as well as products of suitable shifts of the mentioned functions. We consider both archimedean and non-archimedean contribution to the \(\tau\)-Li coefficients, both separately, and their joint contribution to the coefficients. We also derive the behavior of the coefficients in the case the \(\tau/2\)-Riemann hypothesis holds, which is the generalization of the Riemann hypothesis for the class under consideration. Finally, we conclude with some examples and numerics.

### MSC:

11M41 | Other Dirichlet series and zeta functions |

11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |

### Software:

Arb
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\textit{A.-M. Ernvall-Hytönen} et al., Taiwanese J. Math. 22, No. 6, 1321--1346 (2018; Zbl 1429.11172)

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