## Sunday, July 24, 2022

### HP Prime: Curve Fitting to Approximate the Zeta Function

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;

Coming up:  Python Week:  August 1 to August 5, 2022

Next Post:  August 2, 2022

Eddie

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