|
%I #25 Jun 13 2021 05:07:14
%S 9,6,9,2,7,6,9,4,3,8,2,7,4,9,1,6,3,0,7,1,6,9,5,3,7,1,4,7,2,0,9,0,7,3,
%T 2,2,6,6,2,1,3,6,8,8,6,3,8,4,9,1,6,2,1,8,1,6,1,7,8,5,8,8,7,5,1,9,5,0,
%U 5,7,0,0,2,8,3,8,7,4,0,1,9,7,3,4,7,7,8,6,5,0,8,3,3,7,3,4,2,7,6,6,5,0,9,4,8,9
%N Decimal expansion of h = Product_{p prime}(sqrt(p(p-1))*log(1/(1-1/p))).
%C Arises in formulas like: Sum_{k<=x} 1/tau(kd) = hx/sqrt(Pi*log(x))*{ g(d)+O((3/4)^omega(d)/log(x)) } where g satisfies Sum_{d<=x} g(d))=x/h/sqrt(Pi*log(x))*{ 1+O(1/log(x)) }.
%C The logarithm of the value has an expansion -P(2)/24 -P(3)/24 -109*P(4)/2880 -49*P(5)/1440-... in terms of the prime zeta functions P(.). - _R. J. Mathar_, Jan 31 2009
%C The average order of 1/tau(k) (where tau(k) is the number of divisors of k, A000005), Sum_{k<=x} 1/tau(k) ~ h*x/sqrt(Pi*log(x)), was found by Ramanujan in 1916 and was proven by Wilson in 1923. - _Amiram Eldar_, Jun 19 2019
%D G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, collection SMF no. 1, 1995, p. 210.
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.pdf">Some formulae in the analytic theory of numbers</a>, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
%H B. M. Wilson, <a href="https://doi.org/10.1112/plms/s2-21.1.235">Proofs of some formulae enunciated by Ramanujan</a>, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 235-255.
%F Equals A345231 * sqrt(Pi). - _Vaclav Kotesovec_, Jun 13 2021
%e 0.96927694382749163071695371472090732266213688638491621816178588751950570028...
%t $MaxExtraPrecision = 1000; m = 1000; f[p_] := Sqrt[p*(p - 1)]*Log[p/(p - 1)]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* _Amiram Eldar_, Jun 19 2019 *)
%o (PARI) prod(k=1,40000,sqrt(prime(k)*(prime(k)-1))*log(1/(1-1/prime(k))))
%Y Cf. A000005, A345231, A345288.
%K cons,nonn
%O 0,1
%A _Benoit Cloitre_, Jun 02 2003
%E 10 more digits from _R. J. Mathar_, Jan 31 2009
%E More terms from _Amiram Eldar_, Jun 19 2019
%E More digits from _Vaclav Kotesovec_, Jun 13 2021
|
