an:06700094
Zbl 1391.11101
Bucur, Alina; Ernvall-Hyt??nen, Anne-Maria; Od??ak, Almasa; Smajlovi??, Lejla
On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
EN
LMS J. Comput. Math. 19, No. 1, 259-280 (2016).
00352469
2016
j
11M26 11M36 11M41
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.