(* List of fast Lagarias-Miller-Odlyzko alpha factors found by running pi(x) benchmarks. *)

alphaLMO = {(* {x, alpha} *) {1, 1}, {10^10, 1.410}, {10^11, 1.499},  {10^12,1.597},{10^13, 1.681},{10^14, 1.856},{10^15,2.027},{10^16, 2.152},{10^17, 2.329},{10^18, 2.556}}

ListLogLinearPlot[alphaLMO,Filling->Bottom, Joined-> True]

(* alpha is a tuning factor that balances the compuation of the easy special leaves and the hard special leaves. The formula below is used in the file src/common.cpp to calculate a fast alpha factor for the computation of pi(x). *)

NonlinearModelFit[alphaLMO, a(Log[x])^2 + b  Log[x] + c, {a,b, c}, x]