A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

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

Offset

0,1

Comments

The asymptotic density of the numbers k such that k^2+1 is squarefree (A049533) (Estermann, 1931).

The constant c in Sum_{k=0..n} phi(k^2 + 1) = A333170(n) ~ (1/4)*c*n^3 (Finch, 2018).

The constant c in Sum_{k=0..n} phi(k^2 + 1)/(k^2 + 1) = (3/4)*c*n + O(log(n)^2) (Postnikov, 1988).

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 101.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 166.

A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, pp. 192-195.

Links

Table of n, a(n) for n=0..105.

Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Mathematische Annalen, Vol. 105 (1931), pp. 653-662, alternative link.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 14.

D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica, Vol. 155, No. 1 (2012), pp. 1-13.

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], variable F(m=4,n=1,s=2), p. 38.

Wolfgang Schwarz, Über die Summe Sigma_{n <= x} phi(f)(n) und verwandte Probleme, Monatshefte für Mathematik, Vol. 66, No. 1 (1962), pp. 43-54, alternative link.

Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222.

Formula

Equals 2*A065474/A340617.

Example

0.89484122456248817072566150690843732198754780892071...

Maple

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
pzetamod41 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(alfa((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
evalf(exp(-Sum(2^t*pzetamod41(2*t, 50)/t, t = 1..200))); # Vaclav Kotesovec, Jan 13 2021

Mathematica

f[p_] := If[Mod[p, 4] == 1, 1 - 2/p^2, 1]; RealDigits[N[Product[f[Prime[i]], {i, 1, 10^6}], 10], 10, 8][[1]] (* for calculating only the first few terms *)
(* -------------------------------------------------------------------------- *)
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}];
Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Prog

(PARI) f(lim, poly=1-'x-'x^2/2)=prodeulerrat(subst(poly, 'x, 1/'x^2))*prodeuler(p=2, lim, my(pm2=1./p^2); if(p%4==1, 1.-2*pm2, 1.)/subst(poly, 'x, pm2)) \\ Gets 14 digits at lim=1e9; Charles R Greathouse IV, Aug 10 2022

Crossrefs

Cf. A002144, A049533, A065474, A069987, A088539, A333169, A333170, A340617.

Sequence in context: A244664 A010532 A317864 * A243267 A243268 A203071

Adjacent sequences: A335960 A335961 A335962 * A335964 A335965 A335966

Keyword

nonn,cons

Author

Amiram Eldar, Jul 01 2020

Extensions

More digits (from the paper by R. J. Mathar) added by Jon E. Schoenfield, Jan 12 2021

More digits from Vaclav Kotesovec, Jan 13 2021

Status

approved