|
| |
|
|
A337608
|
|
Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.
|
|
2
|
|
|
|
7, 9, 2, 2, 0, 8, 2, 3, 8, 1, 6, 7, 5, 4, 1, 6, 6, 8, 7, 7, 5, 4, 5, 5, 5, 6, 6, 5, 7, 9, 0, 2, 4, 1, 0, 1, 1, 2, 8, 9, 3, 2, 2, 5, 0, 9, 8, 6, 2, 2, 1, 1, 1, 7, 2, 2, 7, 9, 7, 3, 4, 5, 2, 5, 6, 9, 5, 1, 4, 1, 5, 4, 9, 4, 4, 1, 2, 4, 9, 0, 6, 6, 0, 2, 9, 5, 3, 8, 8, 3, 9, 8, 0, 2, 7, 5, 2, 9, 2, 7, 8, 7, 3, 9, 7, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Shanks (1967) conjectured that the number of primes of the form (m + 1)^4 + 1 such that (m - 1)^4 + 1 is also a prime (A217795 plus 1), with m <= x, is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section, evaluated it by 0.79220 and named it after the mathematician Mohan Lal who conjectured the asymptotic formula without evaluating this constant.
The first 100 digits of this constant were calculated by Ettahri et al. (2019).
|
|
|
REFERENCES
|
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 90-91.
|
|
|
LINKS
|
Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
|
|
|
FORMULA
|
Equals (Pi^4/(2^7 * log(1+sqrt(2))^2)) * Product_{primes p == 1 (mod 8)} (1 - 4/p)^2 * ((p + 1)/(p - 1))^4 * p*(p-8)/(p-4)^2 = (Pi^2/32) * A088367^2 * A334826^2 * A210630 = 2 * A337607^2 * A210630.
|
|
|
EXAMPLE
|
0.792208238167541668775455566579024101128932250986221...
|
|
|
MATHEMATICA
|
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
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}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17] * Pi^4/(2^7 * Log[1+Sqrt[2]]^2) * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
