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A257098 From square root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose square is 1/zeta; sequence gives numerator of b(n).

%I A257098 #29 May 05 2025 09:58:02
%S A257098 1,-1,-1,-1,-1,1,-1,-1,-1,1,-1,1,-1,1,1,-5,-1,1,-1,1,1,1,-1,1,-1,1,-1,
%T A257098 1,-1,-1,-1,-7,1,1,1,1,-1,1,1,1,-1,-1,-1,1,1,1,-1,5,-1,1,1,1,-1,1,1,1,
%U A257098 1,1,-1,-1,-1,1,1,-21,1,-1,-1,1,1,-1,-1,1,-1,1,1,1,1,-1,-1,5,-5,1,-1,-1,1,1,1,1,-1,-1,1,1,1,1,1,7,-1,1,1,1
%N A257098 From square root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose square is 1/zeta; sequence gives numerator of b(n).
%C A257098 Dirichlet g.f. of b(n) = A257098(n)/A046644(n) is (zeta(x))^(-1/2).
%C A257098 Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/2).
%C A257098 Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...
%C A257098 The sequence of rationals a(n)/A046644(n) is the Moebius transform of A046643/A046644 which is multiplicative. This sequence is then also multiplicative. - _Andrew Howroyd_, Aug 08 2018
%H A257098 Vaclav Kotesovec, <a href="/A257098/b257098.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Wolfgang Hintze)
%H A257098 Vaclav Kotesovec, <a href="/A257098/a257098.jpg">Graph - the asymptotic ratio (10^7 terms)</a>
%F A257098 with k = 2;
%F A257098 zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
%F A257098 c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
%F A257098 Then solve c(k,n) = mu(n) for b(m);
%F A257098 a(n) = numerator(b(n)).
%F A257098 Sum_{j=1..n} A257098(j)/A046644(j) ~ -n / (2 * sqrt(Pi) * log(n)^(3/2)) * (1 + 3*(gamma/2 + 1)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 05 2025
%t A257098 k = 2;
%t A257098 c[1, n_] = b[n];
%t A257098 c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
%t A257098 nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
%t A257098 sol = Solve[Join[{b[1]==1}, eqs], Table[b[i], {i, 1, nn}], Reals];
%t A257098 t = Table[b[n], {n, 1, nn}] /. sol[[1]];
%t A257098 num = Numerator[t] (* A257098 *)
%t A257098 den = Denominator[t] (* A046644 *)
%o A257098 (PARI) \\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
%o A257098 DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}
%o A257098 apply(numerator, DirSqrt(vector(100, n, moebius(n)))) \\ _Andrew Howroyd_, Aug 08 2018
%o A257098 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%Y A257098 Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).
%Y A257098 Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).
%K A257098 sign,mult,frac
%O A257098 1,16
%A A257098 _Wolfgang Hintze_, Apr 16 2015