A083281 Decimal expansion of h = Product_{p prime}(sqrt(p(p-1))*log(1/(1-1/p))).

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, 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, 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

Offset: 0

Author

Benoit Cloitre, Jun 02 2003

Keywords

• cons
• nonn

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

Examples

			0.96927694382749163071695371472090732266213688638491621816178588751950570028...
		

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.

Crossrefs

Cf. A000005, A345231, A345288.

Programs

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 *)

PARI

prod(k=1,40000,sqrt(prime(k)*(prime(k)-1))*log(1/(1-1/prime(k))))

Formula

Equals A345231 * sqrt(Pi). - Vaclav Kotesovec, Jun 13 2021

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