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On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients. (English) Zbl 1391.11101
Summary: The Li coefficients \(\lambda_F(n)\) of a zeta or \(L\)-function \(F\) provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the \(\tau\)-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport-Heilbronn zeta function. The behavior of the \(\tau\)-Li coefficients varies depending on whether the function in question has any zeros in the half-plane \(\text{Re}(z)>\tau/2.\) We investigate analytically and numerically the behavior of these coefficients for such functions in both the \(n\) and \(\tau\) aspects.

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
Full Text: DOI
[1] DOI: 10.1112/jlms/s1-11.4.307 · Zbl 0015.19802 · doi:10.1112/jlms/s1-11.4.307
[2] DOI: 10.1007/978-3-319-17987-2_7 · Zbl 1383.11065 · doi:10.1007/978-3-319-17987-2_7
[3] DOI: 10.1006/jnth.1999.2392 · Zbl 0972.11079 · doi:10.1006/jnth.1999.2392
[4] DOI: 10.4213/rm9410 · doi:10.4213/rm9410
[5] DOI: 10.1090/S0025-5718-07-01999-0 · Zbl 1130.11046 · doi:10.1090/S0025-5718-07-01999-0
[6] Titchmarsh, The theory of the Riemann zeta-function (1951) · Zbl 0042.07901
[7] DOI: 10.1016/j.jnt.2009.10.012 · Zbl 1188.11046 · doi:10.1016/j.jnt.2009.10.012
[8] Selberg, Proceedings of Amalfi Conference on Analytic Number Theory pp 367– (1992)
[9] DOI: 10.1090/S1061-0022-2013-01242-8 · Zbl 1295.11039 · doi:10.1090/S1061-0022-2013-01242-8
[10] DOI: 10.1112/S1461157010000215 · Zbl 1294.11144 · doi:10.1112/S1461157010000215
[11] Mazhouda, Rocky Mountain J. Math.,
[12] Maslanka, Opuscula Math. 24 pp 103– (2004)
[13] DOI: 10.5802/aif.2311 · Zbl 1216.11078 · doi:10.5802/aif.2311
[14] DOI: 10.1007/BF02392574 · Zbl 1126.11335 · doi:10.1007/BF02392574
[15] Kaczorowski, Analytic number theory, C.I.M.E. Summer School, Cetraro, Italy, 2002 pp 133– (2006)
[16] DOI: 10.1145/2576802.2576828 · Zbl 06408555 · doi:10.1145/2576802.2576828
[17] DOI: 10.1016/j.jnt.2015.03.019 · Zbl 1347.11063 · doi:10.1016/j.jnt.2015.03.019
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