HP Prime: Curve Fitting to Approximate the Zeta Function
Introduction
Here are three approximations for the zeta functions for the positive real numbers x. For the test data, I used the interval 2 ≤ x ≤ 12.
For the even integers, exact values are given, otherwise decimal approximations are given.
2, ζ(2) = π^2 / 6
3, ζ(3) ≈ 1.202056903
4, ζ(4) = π^4 / 90
5, ζ(5) ≈ 1.036927755
6, ζ(6) = π^6 / 945
7, ζ(7) ≈ 1.008349277
8, ζ(8) = π^8 / 9450
9, ζ(9) ≈ 1.002008392
10, ζ(10) = π^10 / 93555
11, ζ(11) ≈ 1.000494189
12, ζ(12) = 691 * π^12 / 638512875
For x → ∞, ζ → 1
Here are results from three curve fits. I have tried to include curve fits of at least 10^-2.
Inverse Regression: Y = A / X + B
Y = 1.42232589936/X+0.81893671619
Average Absolute Error: 5.49240669397ᴇ−2
Logistic Regression: Y = A / (1 - B * (e^(C * X))
Y = 1.00164385688/(1-2.09727867903*e^(-0.839946048322*X))
Average Absolute Error: 1.41745186091ᴇ−3
Custom Regression: Y = A + B / X + C X + D X^2
Y = -0.269041227527+(3.20690850188/X)+0.163810293025*X-6.77810226165ᴇ−3*X^2
Average Absolute Error: 1.05418780589ᴇ−2
HP Prime Program:
EXPORT zetamatrix()
BEGIN
LOCAL R,C;
M1:=MAKEMAT(1,11,4);
M2:=MAKEMAT(approx(CAS.Zeta(I+1)),11,1);
FOR R FROM 1 TO 11 DO
M1(R,2):=approx(1/(R+1));
M1(R,3):=approx(R+1);
M1(R,4):=approx((R+1)^2);
END;
END;
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