|
| |
|
|
A343627
|
|
Decimal expansion of the Prime Zeta modulo function P_{3,1}(7) = Sum 1/p^7 over primes p == 1 (mod 3).
|
|
2
|
|
|
|
0, 0, 0, 0, 0, 1, 2, 3, 1, 3, 7, 2, 2, 5, 5, 4, 8, 1, 9, 1, 9, 6, 7, 4, 4, 4, 8, 9, 4, 7, 1, 2, 4, 4, 4, 4, 0, 0, 3, 9, 3, 6, 6, 6, 9, 0, 5, 7, 8, 6, 6, 2, 6, 3, 7, 0, 7, 2, 8, 1, 9, 6, 3, 7, 0, 6, 2, 0, 2, 1, 0, 5, 7, 4, 1, 2, 0, 6, 7, 2, 6, 0, 0, 6, 9, 5, 5, 9, 2, 2, 1, 2, 7, 4, 9, 2, 4, 8, 2, 5
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
COMMENTS
|
The Prime Zeta modulo function at 7 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^7 = 1/7^7 + 1/13^7 + 1/19^7 + 1/31^7 + ...
The complementary Sum_{primes in A003627} 1/p^7 is given by P_{3,2}(7) = A085967 - 1/3^7 - (this value here) = 0.0078253541130504928742517... = A343607.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
P_{3,1}(7) = 1.231372255481919674448947124444003936669057866...*10^-6
|
|
|
MATHEMATICA
|
With[{s=7}, 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}]] (* adopted from Vaclav Kotesovec's code in A175645 *)
|
|
|
PROG
|
(PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^7); s \\ For illustration: primes up to 10^N give 6N+2 (= 50 for N=8) correct digits.
(PARI) A343627_upto(N=100)={localprec(N+5); digits((PrimeZeta31(7)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
