login
A084892
Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).
6
1, 4, 6, 4, 7, 5, 6, 6, 3, 0, 1, 6, 3, 8, 3, 1, 1, 3, 1, 6, 9, 9, 9, 7, 6, 0, 9, 1, 2, 2, 0, 4, 2, 1, 9, 2, 6, 3, 8, 1, 1, 7, 3, 0, 3, 4, 7, 9, 6, 9, 6, 0, 2, 5, 1, 6, 9, 2, 6, 9, 3, 9, 7, 5, 2, 0, 1, 2, 7, 5, 7, 9, 1, 0, 4, 4, 9, 2, 6, 3, 5, 2, 5, 2, 9, 1, 8, 1, 7, 4, 2, 3, 5, 1, 0, 2, 2, 7, 0, 9, 4, 1
OFFSET
2,2
COMMENTS
This constant, A_2, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_3 = A084893. - Amiram Eldar, Oct 16 2020
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.
LINKS
B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.
EXAMPLE
-14.64756630163831131699976...
MATHEMATICA
m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/2]*Product[Zeta[j/2], {j, 3, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)
PROG
(PARI) prodinf(k=1, if (k!=2, zeta(k/2), 1)) \\ Michel Marcus, Oct 16 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jun 10 2003
STATUS
approved