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

Author

Artur Jasinski, Apr 27 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

			0.03716734354...
		

Crossrefs

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

Programs

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

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].