×

zbMATH — the first resource for mathematics

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.

MSC:
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
Software:
Arb
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.