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]]
%% - %
https://www.math.ucla.edu/~robjohn/math/mathjax.html
Dirichlet divisor problem related sequence.
For n>1 the following two formulas for a(n):
are equivalent and yield sequence https://oeis.org/A368592 starting:
-1, 0, 7, 190, 5826, 214956, 11104542, 711175536, 59256152496, 5925678248160, 730285755406560, 105161159860398720, 18003044434808914560, 3528596711774282883840, 801568243461355261718400, 205201470326854119387494400, 59742508072063053997776844800,...
substituting n! with n, in the second formula for a(n), call it b(n):
yields the sequence:
https://oeis.org/A161664
which is the Dirichlet divisor problem sequence minus the triangular numbers, starting:
0, 0, 1, 2, 5, 7, 12, 16, 22, 28, 37, 43, 54, 64, 75, 86, 101, 113, 130, 144, 161, 179, 200, 216, 238, 260, 283, 305, 332, 354, 383, 409, 438, 468, 499, 526, 561, 595,...
whereas the Dirichlet divisor problem sequence is:
https://oeis.org/A006218
starting:
0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, 113, 119,...
https://or.stackexchange.com/q/10767
n = 4; MatrixForm[Table[Table[Table[ 1, {k, 1 + Floor[h/(m + 1)], Floor[h/m - 1/m]}], {m, 1, 1 + 2*Floor[n/2 - 1/2]}], {h, 1, n!}]]
Table[-1/4*n!*(2 + n!)*(-2 + 1/(1 + Floor[n/2 - 1/2])) - n!*Sum[1/m, {m, 1, 1 + 2*Floor[n/2 - 1/2]}], {n, 1, 17}]