Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH

We have recently proved several explicit versions of the prime ideal theorem under GRH. Here we further explore the method, in order to deduce its strongest consequence for the case where x diverges.

[1]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[2]  A. Odlyzko.  Some analytic estimates of class numbers and discriminants , 1975 .

[3]  J. Barkley Rosser,et al.  Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$ , 1975 .

[4]  Andrew Odlyzko,et al.  Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results , 1990 .

[5]  S. Lang.  Algebraic Number Theory , 1971 .

[6]  Timothy S. Trudgian,et al.  An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions , 2012, Math. Comput..

[7]  Loïc Grenié,et al.  Explicit smoothed prime ideals theorems under GRH , 2013, Math. Comput..

[8]  A. Odlyzko.  Lower bounds for discriminants of number fields , 1976 .

[9]  Vijaya Kumar Murty,et al.  EFFECTIVE VERSIONS OF THE CHEBOTAREV DENSITY THEOREM FOR FUNCTION FIELDS , 1994 .

[10]  H. M. Stark,et al.  Some effective cases of the Brauer-Siegel Theorem , 1974 .

[11]  Bruno Winckler.  Théorème de Chebotarev effectif , 2013 .

[12]  A. Ingham.  The distribution of prime numbers , 1972 .

[13]  H. Kadiri,et al.  Explicit zero density theorems for Dedekind zeta functions , 2012, 1201.3932.

[14]  Lowell Schoenfeld,et al.  Sharper bounds for the Chebyshev functions () and (). II , 1976 .

[15]  Lowell Schoenfeld,et al.  Corrigendum: “Sharper bounds for the Chebyshev functions () and (). II” (Math. Comput. 30 (1976), no. 134, 337–360) , 1976 .

[16]  A. Odlyzko.  Lower bounds for discriminants of number fields, II , 1977 .

[17]  J. Barkley Rosser,et al.  Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II , 1975 .