A382637 Decimal expansion of the multiple prime zeta value p[3, 1].

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

			0.03053116405794...
		

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382636, A383432.

Programs

Mathematica

p3 = N[PrimeZetaP[3], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[
 Do[sum = sum + N[1/p^3, 50]; diff = p3 - sum; sum1 = sum1 + diff/p;
  p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

PARI

f(e)=my(S=sumeulerrat(1/x^3), u=0., v=0); forprime(p=2, 2^e, u+=v*S; S-=1/p^3; v=1/p); u;f(30) // Bill Allombert

Formula

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

p[3, 1] + p[4] = lim_{x->oo} p[3]*p[1](x) - p[1, 3](x) = 0.1075243038221868... = cons, where p[1] and p[1, 3] are divergent series.

p[3, 1] + p[2, 2] + 2 p[2, 1, 1] = lim_{x->oo} p[2, 1]*p[1](x) - p[1, 2, 1](x) = 0.1686331457227234... = cons, where p[1] and p[2, 1, 1] are divergent series.