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# User:Mats Granvik

${\displaystyle 1=1}$
${\displaystyle 1=(1+1)-1}$
${\displaystyle 1=(1+1+1)-(1+1)}$
${\displaystyle 1=(1+1+1+1)-(1+1+1)}$
${\displaystyle 1=(1+1+1+1+...+1)-(1+1+1+...+1)}$
${\displaystyle 1=\underbrace {1} _{n}-\underbrace {1} _{n-1}}$
${\displaystyle T={\begin{pmatrix}1&0&0&0&0&0&0&\cdots \\1&1&0&0&0&0&0\\1&1&1&0&0&0&0\\1&1&1&1&0&0&0\\1&1&1&1&1&0&0\\1&1&1&1&1&1&0\\1&1&1&1&1&1&1\\\vdots &&&&&&&\ddots \end{pmatrix}}}$
${\displaystyle T(n,1)=1}$
${\displaystyle T(n,k)=\left[n>=k\right]\left(\sum _{i=1}^{n-1}T(n-i,k-1)-\sum _{i=1}^{n-1}T(n-i,k)\right)}$

(*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*)

${\displaystyle {\begin{pmatrix}+1&+1&+1&+1&+1&+1&+1&\cdots \\+1&+0&+0&+0&+0&+0&+0\\+1&+0&+0&+0&+0&+0&+0\\+1&+0&+0&+0&+0&+0&+0\\+1&+0&+0&+0&+0&+0&+0\\+1&+0&+0&+0&+0&+0&+0\\+1&+0&+0&+0&+0&+0&+0\\\vdots &&&&&&&\ddots \end{pmatrix}}}$
${\displaystyle {\begin{pmatrix}+1&+1&+1&+1&+1&+1&+1&\cdots \\+1&-1&+0&+0&+0&+0&+0\\+1&+1&+0&+0&+0&+0&+0\\+1&-1&+0&+0&+0&+0&+0\\+1&+1&+0&+0&+0&+0&+0\\+1&-1&+0&+0&+0&+0&+0\\+1&+1&+0&+0&+0&+0&+0\\\vdots &&&&&&&\ddots \end{pmatrix}}}$
${\displaystyle {\begin{pmatrix}+1&+1&+1&+1&+1&+1&+1&\cdots \\+1&-1&+1&-1&+1&-1&+1\\+1&+1&+0&+0&+0&+0&+0\\+1&-1&+0&+0&+0&+0&+0\\+1&+1&+0&+0&+0&+0&+0\\+1&-1&+0&+0&+0&+0&+0\\+1&+1&+0&+0&+0&+0&+0\\\vdots &&&&&&&\ddots \end{pmatrix}}}$
${\displaystyle {\begin{pmatrix}+1&+1&+1&+1&+1&+1&+1&\cdots \\+1&-1&+1&-1&+1&-1&+1\\+1&+1&-2&+1&+1&-2&+1\\+1&-1&+1&-1&+1&-1&+1\\+1&+1&+1&+1&-4&+1&+1\\+1&-1&-2&-1&+1&+2&+1\\+1&+1&+1&+1&+1&+1&-6\\\vdots &&&&&&&\ddots \end{pmatrix}}}$
${\displaystyle T(n,k)}$ = https://oeis.org/A191898 :${\displaystyle (n,k)}$
${\displaystyle \Lambda (n)=\lim \limits _{s\rightarrow 1}\zeta (s)\sum \limits _{d|n}{\frac {\mu (d)}{d^{(s-1)}}}}$
${\displaystyle \sum \limits _{k=1}^{\infty }\sum \limits _{n=1}^{\infty }{\frac {T(n,k)}{n^{c}\cdot k^{s}}}=\sum \limits _{n=1}^{\infty }{\frac {\lim \limits _{z\rightarrow s}\zeta (z)\sum \limits _{d|n}{\frac {\mu (d)}{d^{(z-1)}}}}{n^{c}}}={\frac {\zeta (s)\cdot \zeta (c)}{\zeta (c+s-1)}}}$
${\displaystyle -{\frac {\zeta '(s)}{\zeta (s)}}=\lim _{c\to 1}\,\left({\frac {\zeta (c)\zeta (s)}{\zeta (c+s-1)}}-\zeta (c)\right)}$
${\displaystyle \mu (n)={\underset {n=1}{1}}-{\underset {a=n}{\sum _{a\geq 2}}}1+{\underset {ab=n}{\sum _{a\geq 2}\sum _{b\geq 2}}}1-{\underset {abc=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}}}1+{\underset {abcd=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1-\cdots }$
${\displaystyle 1/a^{b+ic}=1/a^{b}(\cos(c\log(1/a))+i\sin(c\log(1/a)))}$
   1/a^(b + I*c) = 1/a^b*(Cos[c*Log[1/a]] + I*Sin[c*Log[1/a]])

${\displaystyle s=14i}$
${\displaystyle n=7}$
${\displaystyle s-n\left(\lim _{c\to 1}\,{\frac {\sum _{k=1}^{n}{\frac {(-1)^{k-1}{\binom {n-1}{k-1}}}{\zeta ((c-1)(k-1)+s)}}}{\zeta (c)\sum _{k=1}^{n+1}{\frac {(-1)^{k-1}{\binom {n}{k-1}}}{\zeta ((c-1)(k-1)+s)}}}}\right)}$

= 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}]


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

von Mangoldt function matrix:

Divisibility:

   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]


${\displaystyle \Omega (n)={\text{sgn}}\left({\underset {a=n}{\sum _{a\geq 2}}}1\right)+{\text{sgn}}\left({\underset {ab=n}{\sum _{a\geq 2}\sum _{b\geq 2}}}1\right)+{\text{sgn}}\left({\underset {abc=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}}}1\right)+{\text{sgn}}\left({\underset {abcd=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1\right)+\cdots }$

${\displaystyle L(n)={\underset {a^{2}\leq n}{\sum _{a\geq 1}}}1-{\underset {a^{2}b\leq n}{\sum _{a\geq 1}\sum _{b\geq 2}}}1+{\underset {a^{2}bc\leq n}{\sum _{a\geq 1}\sum _{b\geq 2}\sum _{c\geq 2}}}1-{\underset {a^{2}bcd\leq n}{\sum _{a\geq 1}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1+\cdots }$

    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]]
%% - %