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Engineer. $T(n,k)$ = $(n,k)$ $$Lambda(n)=limlimits_{s rightarrow 1} zeta(s)sumlimits_{d n} frac{mu(d)}{d^{(s-1)}}$$ $$sumlimits_{k=1}^{infty}sumlimits_{n=1}^{infty} frac{T(n,k)}{n^c cdot k^s} = sumlimits_{n=1}^{infty} frac{limlimits_{z rightarrow s} zeta(z)sumlimits_{d n} frac{mu(d)}{d^{(z-1)}}}{n^c} = frac{zeta(s) cdot zeta(c)}{zeta(c + s - 1)}$$ $$-frac{zeta `(s)}{zeta (s)}=lim_{cto 1} , left(frac{zeta (c) zeta (s)}{zeta (c+s-1)}-zeta (c)right)$$ $$mu(n) = underbrace{underset{1 = n} 1 - underset{a = n}{sum_{a geq 2}} 1 + underset{ab = n}{sum_{a geq 2} sum_{b geq 2}} 1 - underset{abc = n}{sum_{a geq 2} sum_{b geq 2} sum_{c geq 2}} 1 + underset{abcd = n}{sum_{a geq 2} sum_{b geq 2} sum_{c geq 2} sum_{d geq 2}} 1 - cdots}_{text{#alternating sums}>frac{log(n)}{log(2)}}$$ $$1/a^{b+i c}=1/a^b (cos (c log (1/a))+i sin (c log (1/a)))$$ 1/a^(b + I*c) = 1/a^b*(Cos[c*Log[1/a]] + I*Sin[c*Log[1/a]]) $$f(n,s)=frac{(s+1)^{n-1}+s-1}{s}$$ 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}] (*Mathematica start*) x = N[Exp[-ZetaZero[1]/10], 100] Sum[(-1)^k*x^(Log[k]*10), {k, 1, Infinity}] (*end*) =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)))) 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)) Profile ©