}// end int Moebius_Mi( Int64 theIsolatedNatural )
/// <summary>
/// It's an integral function (in the sense of a cumulative function).
/// NB. have to start from (+1) not from zero !!!!
/// </summary>
/// <param name="theIsolatedNatural"></param>
/// <returns></returns>
public static long Mertens_M(Int64 theIsolatedNatural)
{ // It's an integral function (in the sense of a cumulative function).
long Mertens_M = 0L; // sum on it; it's the reduct of a series.
for (int c = 1; c <= theIsolatedNatural; c++) // NB. have to start from (+1) not from zero !!!!
{ // NB. have to start from (+1) not from zero !!!!
Mertens_M += ElementFullCalculation.Moebius_Mi(c);
}// end for : // NB. have to start from (+1) not from zero !!!!
// ready
return(Mertens_M);
}// end long Mertens_M( Int64 theIsolatedNatural )
}// end int DiracDelta_Omegas( Int64 theIsolatedNatural )
public static int Moebius_Mi(Int64 theIsolatedNatural)
{
if (+1L > theIsolatedNatural)
{
throw new System.Exception("Omega functions are defined for n in N, n>=+1.");
}// else can continue.
//
int DiracDelta_onOmegas = ElementFullCalculation.DiracDelta_Omegas(theIsolatedNatural);
int Lambda = ElementFullCalculation.Lambda_Liouville(theIsolatedNatural);
// ready
return(DiracDelta_onOmegas * Lambda);
}// end int Moebius_Mi( Int64 theIsolatedNatural )