(*Definition:*) FrancaLeClair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]; Monitor[f = Table[Sign[Im[ZetaZero[n]] - FrancaLeClair[n]], {n, 1, 90}];, n]; (1 + f)/2 (*Comment 1:*) Monitor[a = Table[Floor[ Im[ZetaZero[n]]/(2*Pi)*Log[Im[ZetaZero[n]]/(2*Pi*Exp[1])] + 11/8 - n + 1], {n, 1, 90}];, n]; a (*Comment 2:*) Monitor[Table[(1 - Sign[Im[Zeta[1/2 + I*2*Pi*E*Exp[LambertW[(n - 11/8)/E]]]]])/ 2, {n, 1, 90}], n] (*Comment 3:*) Table[Floor[ 2*(RiemannSiegelTheta[Im[ZetaZero[n]]]/Pi - Floor[RiemannSiegelTheta[Im[ZetaZero[n]]]/Pi])], {n, 1, 90}] (*Comment 3:*) (*For n>1:*) Floor[2*FractionalPart[ N[RiemannSiegelTheta[Im[ZetaZero[Range[90]]]]/Pi, 30]]] (*Comment 4:*) Clear[nn, n, k, t, FrancaLeclair, NumberOfZetaZeros]; nn = 89; FrancaLeclair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]; NumberOfZetaZeros[t_] = RiemannSiegelTheta[t]/Pi + Im[Log[Zeta[1/2 + I*t]] + I*Pi]/Pi; Monitor[b = N[Table[-(1 + 2*Sum[(NumberOfZetaZeros[FrancaLeclair[k + 1]] - 1) - (NumberOfZetaZeros[FrancaLeclair[k]] - 1) - 1, {k, 1, n}]), {n, 0, nn}]], n]; (*Notice that sequence b is integer in itself despite the use of the \ Round function below*) (1 + Round[b])/2 (*End*)