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User:Mats Granvik

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Engineer. http://mobiusfunction.wordpress.com/ http://www.facebook.com/mats.granvik


(*Mathematica start*)
Clear[t];
nn = 8;
t[n_, 1] = 1;
t[n_, k_] := 
t[n, k] = 
 If[n >= k, 1, 
   0]*(Sum[t[n - i, k - 1], {i, 1, n - 1}] - 
    Sum[t[n - i, k], {i, 1, n - 1}])
MatrixForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]
(*Mathematica end*)
= https://oeis.org/A191898 :
   1/a^(b + I*c) = 1/a^b*(Cos[c*Log[1/a]] + I*Sin[c*Log[1/a]])

= 0.5 + 14.1347i



   N[Table[2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]], {n, 1, 12}]]
   
   Plot[RiemannSiegelTheta[t]/Pi + 
     Im[Log[Zeta[1/2 + I*t]] + I*Pi]/Pi, {t, 0, 60}, ImageSize -> Large]
   Table[Limit[
     Zeta[s] Total[1/Divisors[n]^(s - 1)*MoebiusMu[Divisors[n]]], 
     s -> 1], {n, 1, 32}]

=IF(OR(ROW()=1; COLUMN()=1);0; IF(ROW()>=COLUMN();EXP(-(1-11/8/(COLUMN()-1))/EXP(1)*SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1; COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4)));0))

divided with: /2/PI()/EXP(1) gives reciprocal.

von Mangoldt function matrix:

=IF(OR(ROW()=1, COLUMN()=1), 1, IF(ROW()>=COLUMN(),-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1,COLUMN(), 4)&":"&ADDRESS(ROW()-1, COLUMN(), 4), 4)),-SUM(INDIRECT(ADDRESS(COLUMN()-ROW()+1,ROW(), 4)&":"&ADDRESS(COLUMN()-1, ROW(), 4), 4))))

http://pastebin.com/u/MatsGranvik

Divisibility:

=IF(OR(COLUMN()=1); 1; IF(ROW()>=COLUMN();SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN()-1; 4)&":"&ADDRESS(ROW()-1; COLUMN()-1; 4); 4))-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));0))

   Clear[nn];
   
   nn = 12
   
   f[n_, s_] = ((s + 1)^(n - 1) + s - 1)/s;
   
   TableForm[
    FullSimplify[
     Table[Integrate[Integrate[f[n, s], {n, 1, 2}], {s, 0, k}], {k, 0, 
       nn}]]]
   
   Table[Limit[f[n, s], s -> 0], {n, 1, nn}]
   
   Table[Limit[D[f[n, s], s], s -> 0], {n, 1, nn}]
   
   Table[Limit[Integrate[f[-n, s], s], s -> 0], {n, 1, nn}]
   
   FullSimplify[
    Differences[Table[Limit[Sum[f[-n, s], s], s -> 0], {n, -1, nn}]]]
   
   Table[Limit[(-1 + n s (1 + s) + (2 + s)^n)/((1 + s)^2), s -> -1], {n, 1, nn}]
   
   z = Integrate[((s + 1)^(-n - 1) + s - 1)/s, s];
   a = Limit[z, s -> 0]




    nn = 400;
    a = Table[Sum[If[a == n, 1, 0], {a, 2, n}], {n, 1, nn}]
    b = Table[Sum[If[a^2 == n, 1, 0], {a, 1, n}], {n, 1, nn}]
    c = b;
    T = Table[
       b = Table[
         Sum[If[Mod[n, k] == 0, an/k*bk, 0], {k, 1, n}], {n, 1, 
          nn}], {i, 1, 12}];
    c + Sum[(-1)^n*Tn, All, {n, 1, 12}]
    LiouvilleLambda[Range[nn]]
    %% - %