OFFSET
0,1
COMMENTS
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)) }.
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
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
REFERENCES
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, collection SMF no. 1, 1995, p. 210.
LINKS
S. Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 235-255.
FORMULA
Equals A345231 * sqrt(Pi). - Vaclav Kotesovec, Jun 13 2021
EXAMPLE
0.96927694382749163071695371472090732266213688638491621816178588751950570028...
MATHEMATICA
$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 *)
PROG
(PARI) prod(k=1, 40000, sqrt(prime(k)*(prime(k)-1))*log(1/(1-1/prime(k))))
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jun 02 2003
EXTENSIONS
10 more digits from R. J. Mathar, Jan 31 2009
More terms from Amiram Eldar, Jun 19 2019
More digits from Vaclav Kotesovec, Jun 13 2021
STATUS
approved