A382634 Decimal expansion of the multiple prime zeta value p[2, 3].

0, 2, 9, 1, 8, 5, 1, 6, 6, 5, 0, 4, 0, 1, 2, 5, 7, 1, 0, 4, 0, 6, 4, 1, 3, 4, 4, 0, 9, 0, 2, 7, 9, 1, 9, 6, 7, 4, 7

Offset: 0

Author

Artur Jasinski, Apr 01 2025

Keywords

• nonn
• cons
• more

Comments

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.

For complete list of multiple prime zeta values up to weight 6 see A382234.

Examples

			0.029185166504012571040641344090279196747...
		

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382635.

Cf. A258986.

Programs

Mathematica

p2 = N[PrimeZetaP[2], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[Do[sum = sum + N[1/p^2, 50]; diff = p2 - sum; sum1 = sum1 + diff/p^3; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

Formula

Equals Sum_{p,q prime p>q} 1/(p^2*q^3).

Equals A085548*A085541 - A085965 - A382635.

For partial sums and in infinity occurs identity:

p[2, 3] + p[3, 2] + p[2, 1, 2] + p[2, 2, 1] = p[1] p[2, 2]- p[1, 2, 2]

where p[1] and p[1, 2, 2] are divergent series then

lim_{n->oo} p[1](n)*A382234 - p[1, 2, 2](n) = 0.067101047034256...