A382637 -id:A382637 - cp's OEIS frontend

A382636 Decimal expansion of the multiple prime zeta value p[2, 1].

1, 5, 2, 6, 6, 1, 4, 1, 1, 2, 5, 4, 2

Offset: 0

Artur Jasinski, Apr 07 2025

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.

			0.1526614112542...

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

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; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

Equals p[2, 1] = Sum_{p,q prime p>q} 1/(p^2*q).
Equals p[2, 1] = (p[2, 3] + p[4, 1] + p[2, 1, 2] + 2 p[2, 2, 1])/A085548.
Equals p[2, 1] = sqrt(p[4, 2] + 2 p[2, 2, 2] + 2 p[2, 3, 1] + 2 p[4, 1, 1] + 2 p[2, 1, 2, 1] + 4 p[2, 2, 1, 1]).
A085548*p[2, 1] - p[2, 1, 2] = 0.0531558219243989116479829... [25 digits accuracy].
For partial sums and in infinity occurs identities:
(1) lim_{x->oo} (p[1](x)*A085548 - p[1, 2](x)) = p[2, 1] + A085541 = const.
(2) lim_{x->oo} p[1](x)^3 - 2*p[1](x)*A085548 - p[1, 2](x) - 6*p[1, 1, 1](x) = p[2, 1] - A085541 = const.
(3) lim_{x->oo} (p[1](x)^3 - 3*p[1, 2](x) - 6*p[1, 1, 1](x) = 3*p[2, 1] + A085541 = const.
(4) lim_{x->oo} (p[1](x)^3 - p[1](x)*A085548 - p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
(5) lim_{x->oo} (p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
on the left side of each eq. (1)-(5) are divergent series: p[1], p[1, 1], p[1, 2], p[1, 1, 1].

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

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Apr 27 2025

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.

			0.03716734354...

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

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

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].

cp's OEIS Frontend

A382636 Decimal expansion of the multiple prime zeta value p[2, 1].

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