I've been experimenting with the convergence acceleration of alternating series of the eta formula,
,
where B is the MRB constant and eta^[m] is the m'th derivative DirichletEta function.
I came up with a couple of programs:
One called m1, which with an call the the Sum command gives an approximation of the MRB constant.
m1[u_, p_] :=
Module[{s, n, d, a, b, c}, n = Floor[1.5 p]; d = (3 + Sqrt[8])^n;
d = 1/2*(d + 1/d);
{b, c, s} = {-1, -d, 0};
Do[c = b - c;
a = (Log[k + 1]/(k + 1))^u/u!;
s = s + c*a;
b = (k + n)*(k - n)*b/((k + 1)*(k + 1/2)), {k, 0, n}];
(N[-s/d, n/1.5])]
followed by
p = 100; Sum[m1[u, p], {u, 1, Floor[p/1.8]}]
gives 100 or p digits.
The second one, called meta1, gives an arithmetic sequence for exactly, or more than, p correct digits of the the alternating sum of "ets" eta derivatives shown here:
m2[u_, p_] :=
Module[{s, n, d, a, b, c}, n = Floor[1.5 p]; d = Cos[n ArcCos[3]];
{b, c, s} = {-1, -d, 0};
Do[c = b - c;
a = (Log[k + 1]/(k + 1))^u/u!;
s = s + c*a;
b = (k + n)*(k - n)*b/((k + 1)*(k + 1/2)), {k, 0, n}];
(-s/d)]
followed by
ets = 1; p = 10; meta1 = Sum[m2[u, p], {u, 1, ets}]
,
gives the new, "trig-log," sequence that follows.
I
(-(1/2) (451 - Cos[15 ArcCos[3]]) Log[2] -
1/3 (-34051 + Cos[15 ArcCos[3]]) Log[3] -
1/4 (1024131 - Cos[15 ArcCos[3]]) Log[4] -
1/5 (-16299651 + Cos[15 ArcCos[3]]) Log[5] -
1/6 (158192259 - Cos[15 ArcCos[3]]) Log[6] -
1/7 (-1018147459 + Cos[15 ArcCos[3]]) Log[7] -
1/8 (4590269059 - Cos[15 ArcCos[3]]) Log[8] -
1/9 (-15068492419 + Cos[15 ArcCos[3]]) Log[9] -
1/10 (37120701059 - Cos[15 ArcCos[3]]) Log[10] -
1/11 (-70547206787 + Cos[15 ArcCos[3]]) Log[11] -
1/12 (106723078787 - Cos[15 ArcCos[3]]) Log[12] -
1/13 (-133986054787 + Cos[15 ArcCos[3]]) Log[13] -
1/14 (147575599747 - Cos[15 ArcCos[3]]) Log[14] -
1/15 (-151602131587 + Cos[15 ArcCos[3]]) Log[15] -
1/16 (152139002499 - Cos[15 ArcCos[3]]) Log[16]) Sec[15 ArcCos[3]]
You can check and see it gave more than 10 digits of the first eta derivative:
Limit[DirichletEta'[x], x -> 1] - N[meta1, 20]
-7.576779*10^-14
They both could use some improvement. Maybe someone would like to help me.
I've got to work on my job all day tomorrow, so I wont have any time to work on it myself. Any help here?
More on the convergence acceleration of alternating series is found here:
http://projecteuclid.org/euclid.em/1046889587
I gave it my own trigonometric subroutine for d, which shaves a little time off the computation and gives series in trig functions.DirichletEta' gives results in zeta and zeta^[m] functions. See below.
D[DirichletEta[u], {u, 7}] /. u -> 7
(* (1/64)*Log[2]^7*Zeta[7] - (7/64)*Log[2]^6*
Derivative[1][Zeta][7] +
(21/64)*Log[2]^5*Derivative[2][Zeta][7] -
(35/64)*Log[2]^4*Derivative[3][Zeta][7] +
(35/64)*Log[2]^3*Derivative[4][Zeta][7] -
(21/64)*Log[2]^2*Derivative[5][Zeta][7] +
(7/64)*Log[2]*Derivative[6][Zeta][7] +
(63/64)*Derivative[7][Zeta][7]*)
