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A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1). 15
1, 0, 1, 0, 9, 1, 5, 1, 6, 0, 6, 0, 1, 0, 1, 9, 5, 2, 2, 6, 0, 4, 9, 5, 6, 5, 8, 4, 2, 8, 9, 5, 1, 4, 9, 2, 0, 9, 8, 4, 5, 3, 8, 6, 2, 7, 5, 8, 1, 7, 3, 8, 5, 2, 3, 7, 3, 2, 0, 2, 4, 2, 0, 0, 8, 9, 2, 5, 1, 6, 1, 3, 7, 4, 2, 4, 5, 6, 7, 2, 6, 3, 7, 0, 9, 3, 9, 6, 1, 9, 7, 6, 9, 4, 5, 5, 8, 9, 2, 1, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..501

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2014-2015, Section 3.3. zeta_{5,1}(2).

FORMULA

Equals Sum_{k>=1} 1/A004615(k)^2. - Amiram Eldar, Jan 24 2021

Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - Artur Jasinski, Jan 30 2021

EXAMPLE

1.01091516060101952260495658428951492...

MATHEMATICA

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);

P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];

Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);

$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[5, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021, took 20 minutes *)

CROSSREFS

Cf. A340127, A340628, A340629, A340665, A340794, A340839.

Cf. A175646, A301429, A333240.

Cf. A175647, A248930, A248938, A335963.

Cf. A004615, A340576, A340577, A340578, A334826.

Sequence in context: A298512 A192930 A010168 * A327963 A328112 A328097

Adjacent sequences:  A340001 A340002 A340003 * A340005 A340006 A340007

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 15 2021

STATUS

approved

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Last modified September 21 12:54 EDT 2021. Contains 347598 sequences. (Running on oeis4.)