A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

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

Offset: 0

Author

M. F. Hasler, Apr 25 2021

Keywords

• nonn
• cons

Comments

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

Examples

			0.015689614727130461563527666152209091814208675553077763366153188676457...
		

Links

• R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=6), p. 21.
• OEIS index to entries related to the (prime) zeta function.

Crossrefs

Cf. A003627 (primes 3k-1), A001014 (n^6), A085966 (PrimeZeta(6)), A021733 (1/3^6).

Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).

Cf. A343626 (for primes 3k+1), A086036 (for primes 4k+1), A085995 (for primes 4k+3).

Programs

PARI

A343616_upto(N=100)={localprec(N+5); digits((PrimeZeta32(6)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

Formula

P_{3,2}(6) = Sum_{p in A003627} 1/p^6 = P(6) - 1/3^6 - P_{3,1}(6).