A343619 -id:A343619 - cp's OEIS frontend

A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

			5.121028122527738383259898597063472005396598569391504803757141806973300...* 10^-7

Jean-François Alcover, Table of n, a(n) for n = 0..1009
X. Gourdon and P. Sebah, Some Constants from Number theory.
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=4, n=1, s=9), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085998 (same for primes 4k+3), A343629 (for primes 3k+1), A343619 (for primes 3k+2), A086032 - A086038 (for 1/p^2, ..., 1/p^8), A085969 (PrimeZeta(9)), A002144 (primes of the form 4k+1).

digits = 1004;
nmax0 = 50; (* initial number of sum terms *)
dnmax = 10; (* nmax increment *)
dd = 10; (* precision excess *)
Clear[PrimeZeta41];
f[s_] := (1 + 2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2 s];
PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd]&;
PrimeZeta41[9, nmax = nmax0];
PrimeZeta41[9, nmax += dnmax];
While[Abs[PrimeZeta41[9, nmax] - PrimeZeta41[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta41[9] = PrimeZeta41[9, nmax];
Join[{0, 0, 0, 0, 0, 0}, RealDigits[PrimeZeta41[9], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 07 2021 *)

A086039_upto(N=100)={localprec(N+3); digits((PrimeZeta41(9)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(9) = Sum_{p in A002144} 1/p^9, where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(9m)*zeta(9m)/zeta(18m)/(1 + 2^(-9m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

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.

			0.30792075860773643684250507594099872658103266547551448005201925299378554901...

Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085548 (PrimeZeta(2)), A021031 (1/27).

Cf. A175644 (same for primes 3k+1), A086032 (for primes 4k+1), A085991 (for primes 4k+3), A343613 - A343619 (P_{3,2}(s): same with 1/p^s, s = 3, ..., 9).

digits = 105; nmax0 = 20; dnmax = 5;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = 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, nmax}] // N[#, digits+5]&;
PrimeZeta31[2, nmax = nmax0];
PrimeZeta31[2, nmax += dnmax];
While[Abs[PrimeZeta31[2, nmax] - PrimeZeta31[2, nmax-dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[2] = PrimeZetaP[2] - 1/3^2 - PrimeZeta31[2, nmax];
RealDigits[PrimeZeta32[2], 10, digits][[1]] (* Jean-François Alcover, May 06 2021, after M. F. Hasler's PARI code *)

s=0; forprimestep(p=2,1e8,3,s+=1./p^2);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
PrimeZeta32(s)={sumeulerrat(1/p^s)-1/3^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))}
A343612_upto(N=100)={localprec(N+5); digits(PrimeZeta32(2)\.1^N)}

P_{3,2}(2) = P(2) - 1/3^2 - P_{3,1}(2) = A085548 - A000012 - A175644.

A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

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.

			0.134125178915463540428599329999431198995879919752168337370599106153853349956...

Jean-François 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.

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).

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.

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

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

A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

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.

			P_{3,2}(4) = 0.06418614569655777899009908658740273681...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085964 (PrimeZeta(4)), A021085 (1/3^4).

Cf. A343624 (same for primes 3k+1), A086034 (for primes 4k+1), A085993 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343614_upto(N=100)={localprec(N+5); digits((PrimeZeta32(4)+1)\.1^N)[^1]} \\ see for the function PrimeZeta32.

P_{3,2}(4) = P(4) - 1/3^4 - P_{3,1}(4) = A085964 - A021085 - A343624.

A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

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.

			0.0315771357190039419560337803437163963477729963832486145790258341228297557...

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=5) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A000584 (n^5), A085965 (PrimeZeta(5)), A021247 (1/3^5).

Cf. A343625 (same for primes 3k+1), A086035 (for primes 4k+1), A085994 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343615_upto(N=100, s=5)={localprec(N+5); digits((sumeulerrat(1/p^s)-1/3^s-PrimeZeta31(s)+1)\.1^N)[^1]} \\ see A175644 for the function PrimeZeta31, A343612 for a function PrimeZeta32

P_{3,2}(5) = P(5) - 1/3^5 - P_{3,1}(5).

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

Original entry on oeis.org

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

M. F. Hasler, Apr 25 2021

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.

			0.015689614727130461563527666152209091814208675553077763366153188676457...

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.

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).

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

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

A343618 Decimal expansion of P_{3,2}(8) = Sum 1/p^8 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

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.

			0.003908814823388594971406115663072323981226161069324694978359864189332...

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=8), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001016 (n^8), A085968 (PrimeZeta(8)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343628 (for primes 3k+1), A086038 (for primes 4k+1), A085997 (for primes 4k+3).

A343618_upto(N=100)={localprec(N+5); digits((PrimeZeta32(8)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(8) = Sum_{p in A003627} 1/p^8 = P(8) - 1/3^8 - P_{3,1}(8).

A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

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.

			0.0078253541130504928742517016707559206033079309751324433146804883394...

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=7), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

cp's OEIS Frontend

A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.

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A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

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A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

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A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

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A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

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

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A343618 Decimal expansion of P_{3,2}(8) = Sum 1/p^8 over primes == 2 (mod 3).

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