A383432 Decimal expansion of the multiple prime zeta value p[2, 1, 1].

0, 3, 7, 1, 6, 7, 3, 4, 3, 5, 4

Offset

0,2

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.

Links

Table of n, a(n) for n=0..10.

Formula

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

Equals (p[2, 1, 3] + p[2, 3, 1] + p[4, 1, 1] + p[2, 1, 1, 2] + p[2, 1, 2, 1] + 2 p[2, 2, 1, 1])/p[2].

Equals (p[2] p[4] + p[2, 1]^2 + p[2] p[2, 2] + p[2, 4] + p[2, 1, 3] + p[2, 2, 2] - p[2, 3, 1] - p[4, 1, 1] + p[2, 1, 1, 2] - p[2, 1, 2, 1] - 2 p[2, 2, 1, 1] - p[2]^3)/p[2].

Example

0.03716734354...

Prog

(PARI)
f(e)=my(S=sumeulerrat(1/x^2), u=0., v=0, w=0.); forprime(p=2, prime(2^e), u+=v*S; S-=1/p^2; v=w/p; w+=1/p); u;
f(30) \\ Bill Allombert

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382636, A382637.

Sequence in context: A010781 A019806 A199589 * A354977 A080172 A385485

Adjacent sequences: A383429 A383430 A383431 * A383433 A383434 A383435

Keyword

nonn,cons,more

Author

Artur Jasinski, Apr 27 2025

Status

approved