Dirichlet Eta Function


The Dirichlet Eta function Eta(x) = 1 − 2^−x + 3^−x − ... = Sum{k=1, infinity}[(−1)^(k+1)*(k^−x)], x > 0. Eta is defined for all values of x (even complex x).

For example, Eta(1) = 1 − 1/2 + 1/3 − 1/4 + ... = Ln(2) = 6.93147,18055,99453,09417...E−1.

Here are some notes from my program XPCalc - Extra Precision Floating-Point Calculator http://www.oocities.org/hjsmithh/download.html#XPCalc :

Eta(x) = Dirichlet Eta function:

Dirichlet Eta function of x > 0, is defined by the series 1 − 1/2^x + 1/3^x − 1/4^x + ... .

For of x >= −4, Eta(x) is computed by

Sum{k=1 to n}[(−1)^(k+1) * k^(−x)] + 2^(−n) * Sum{k=n+1 to 2*n}[((−1)^(k+1) * k^(−x) * Sum{j=0 to 2*n−k}[C(n, j)]

where n = (int)(5 + 1.3 * decimal-digits desired) + 5 and C(n, j) is the binomial coefficient.

For negative x < −4, let y = 1−x, then

Eta(x) = 2 * (2*Pi)^y * Sin(Pi*x/2) * Gam(y) * Eta(y) * (1−2^y)/(1−2^x).

Eta(0) = 1/2. Eta(1) = Ln(2). Eta(2) = Pi^2 / 12. Eta(3) = 3*Zeta(3)/4. Eta(4) = 7*Pi^4 / 720. Eta(x) = 0 for all negative even integers.

z = Eta(x) // Dirichlet Eta function for x >= -4 { if (x == 0) return 1/2; if ((x < 0) && (x is even)) return 0; n = (int)(5 + 1.3 * decimal-digits desired) + 5; sum = 0; sign = -1; // (-1)^(k-1) c = 1; // c = C(n, j) e = 1; // e = Sum C(n, j) m = n; // m = n - j + 1 j = 1; for (k = n+n; k > n; k--) { sum += sign * e * (k ^ -x); c = (c * m) / j; e += c; m--; j++; sign = -sign; } sum /= (2 ^ n); for (k = n; k > 0; k--) { sum += sign * (k ^ -x); sign = -sign; } return sum; }

See: Dirichlet Eta Function -- From MathWorld
And: Wolfram Function Evaluation -- Zeta (Eta(x) = (1−2^(1−x))*Zeta(x))

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Changes last made on Sunday, 29-Mar-09 07:18:07 PDT