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A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3). 4
1, 3, 4, 1, 2, 5, 1, 7, 8, 9, 1, 5, 4, 6, 3, 5, 4, 0, 4, 2, 8, 5, 9, 9, 3, 2, 9, 9, 9, 9, 4, 3, 1, 1, 9, 8, 9, 9, 5, 8, 7, 9, 9, 1, 9, 7, 5, 2, 1, 6, 8, 3, 3, 7, 3, 7, 0, 5, 9, 9, 1, 0, 6, 1, 5, 3, 8, 5, 3, 3, 4, 9, 9, 5, 6, 0, 4, 7, 9, 3, 7, 6, 7, 1, 5, 2, 8, 6, 5, 3, 7, 4, 0, 4, 0, 3, 4, 4, 4, 3, 3, 6, 7, 8, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

Jean-Fran├žois Alcover, Table of n, a(n) for n = 0..1006

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=3) on p. 21.

OEIS index to entries related to the (prime) zeta function.

FORMULA

P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.

EXAMPLE

0.134125178915463540428599329999431198995879919752168337370599106153853349956...

PROG

(PARI) s=0; forprimestep(p=2, 1e8, 3, s+=1./p^3); s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.

(PARI) A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.

CROSSREFS

Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).

Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

Sequence in context: A106650 A076942 A205127 * A333454 A021971 A260822

Adjacent sequences:  A343610 A343611 A343612 * A343614 A343615 A343616

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Apr 22 2021

STATUS

approved

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Last modified July 29 06:44 EDT 2021. Contains 346340 sequences. (Running on oeis4.)