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A257099 From third root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is 1/zeta; sequence gives numerator of b(n).

%I A257099 #33 May 05 2025 09:57:34
%S A257099 1,-1,-1,-1,-1,1,-1,-5,-1,1,-1,1,-1,1,1,-10,-1,1,-1,1,1,1,-1,5,-1,1,
%T A257099 -5,1,-1,-1,-1,-22,1,1,1,1,-1,1,1,5,-1,-1,-1,1,1,1,-1,10,-1,1,1,1,-1,
%U A257099 5,1,5,1,1,-1,-1,-1,1,1,-154,1,-1,-1,1,1,-1,-1,5,-1,1,1,1,1,-1,-1,10,-10,1,-1,-1,1,1,1,5,-1,-1,1,1,1,1,1,22,-1,1,1,1
%N A257099 From third root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is 1/zeta; sequence gives numerator of b(n).
%C A257099 Dirichlet g.f. of b(n) = a(n)/A256689(n) is (zeta(x))^(-1/3).
%C A257099 Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/3).
%C A257099 Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...
%H A257099 Vaclav Kotesovec, <a href="/A257099/b257099.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Wolfgang Hintze)
%H A257099 Vaclav Kotesovec, <a href="/A257099/a257099.jpg">Graph - the asymptotic ratio (10^7 terms)</a>
%F A257099 with k = 3;
%F A257099 zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
%F A257099 c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
%F A257099 Then solve c(k,n) = mu(n) for b(m);
%F A257099 a(n) = numerator(b(n)).
%F A257099 Sum_{j=1..n} A257099(j)/A256689(j) ~ n / (Gamma(-1/3) * log(n)^(4/3)) * (1 + 4*(gamma/3 + 1)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - _Vaclav Kotesovec_, May 05 2025
%t A257099 k = 3;
%t A257099 c[1, n_] = b[n];
%t A257099 c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
%t A257099 nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
%t A257099 sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
%t A257099 t = Table[b[n], {n, 1, nn}] /. sol[[1]];
%t A257099 num = Numerator[t] (* A257099 *)
%t A257099 den = Denominator[t] (* A256689 *)
%o A257099 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/3))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%Y A257099 Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).
%Y A257099 Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).
%K A257099 sign,frac
%O A257099 1,8
%A A257099 _Wolfgang Hintze_, Apr 16 2015