OFFSET
0,2
COMMENTS
The prime zeta modulo function at 2 for primes of the form 3k+1, which is P_{3,2}(2) = Sum_{p in A002476} 1/p^2 = 1/7^2 + 1/13^2 + 1/19^2 + 1/31^2 + ...
LINKS
Jean-François Alcover, Table of n, a(n) for n = 0..1003
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
EXAMPLE
P_{3,1}(2) = 0.03321555032221795055292717778013809648108756665...
MATHEMATICA
With[{s=2}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* Vaclav Kotesovec, Jan 13 2021 *)
digits = 1003;
m = 100; (* initial value of n beyond which summand is considered negligible *)
dm = 100; (* increment of m *)
P[s_, m_] (* "P" short name for "PrimeZeta31" *):= P[s, m] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1]* ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, m}] // N[#, digits+10]&;
P[2, m]; P[2, m += dm];
While[ RealDigits[P[2, m]][[1]][[1;; digits]] !=
RealDigits[P[2, m-dm]][[1]][[1;; digits]], Print["m = ", m]; m+=dm];
Join[{0}, RealDigits[P[2, m]][[1]][[1;; digits]]] (* Jean-François Alcover, Jun 24 2022, after Vaclav Kotesovec *)
PROG
(PARI)
my(s=0); forprimestep(p=1, 1e8, 3, s+=1./p^2); s \\ For illustration: primes up to 10^N give only about 2N+2 (= 18 for N=8) correct digits. - M. F. Hasler, Apr 23 2021
(PARI)
PrimeZeta31(s)=suminf(n=0, my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t, 1/6)-zetahurwitz(t, 5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2)) \\ Inspired from Kotesovec's Mmca code
A175644_upto(N=100)={localprec(N+5); digits((PrimeZeta31(2)+1)\.1^N)[^1]} \\ M. F. Hasler, Apr 23 2021
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
R. J. Mathar, Aug 01 2010
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 27 2020
STATUS
approved