Riemann Zeta Function in Java - Infinite Recursion with Functional Form

Question

Note: Updated on 06/17/2015. Of course this is possible. See the solution below.

Even if anyone copies and pastes this code, you still have a lot of cleanup to do. Also note that you will have problems inside the critical strip from Re(s) = 0 to Re(s) = 1 :). But this is a good start.

import java.util.Scanner;

public class NewTest{

public static void main(String[] args) {
    RiemannZetaMain func = new RiemannZetaMain();
    double s = 0;
    double start, stop, totalTime;
    Scanner scan = new Scanner(System.in);
    System.out.print("Enter the value of s inside the Riemann Zeta Function: ");
    try {
            s = scan.nextDouble();
    } 
    catch (Exception e) {
        System.out.println("You must enter a positive integer greater than 1.");
    }
    start = System.currentTimeMillis();
    if (s <= 0)
        System.out.println("Value for the Zeta Function = " + riemannFuncForm(s));
    else if (s == 1)
        System.out.println("The zeta funxtion is undefined for Re(s) = 1.");
    else if(s >= 2)
        System.out.println("Value for the Zeta Function = " + getStandardSum(s));
    else
        System.out.println("Value for the Zeta Function = " + getNewSum(s));
    stop = System.currentTimeMillis();
    totalTime = (double) (stop-start) / 1000.0;
    System.out.println("Total time taken is " + totalTime + " seconds.");
}

// Standard form the the Zeta function.
public static double standardZeta(double s) {
    int n = 1;
    double currentSum = 0;
    double relativeError = 1;
    double error = 0.000001;
    double remainder;

    while (relativeError > error) {
        currentSum = Math.pow(n, -s) + currentSum;
        remainder = 1 / ((s-1)* Math.pow(n, (s-1)));
        relativeError =  remainder / currentSum;
        n++;
    }
    System.out.println("The number of terms summed was " + n + ".");
    return currentSum;
}

public static double getStandardSum(double s){
    return standardZeta(s);
}

//New Form
// zeta(s) = 2^(-1+2 s)/((-2+2^s) Gamma(1+s)) integral_0^infinity t^s sech^2(t) dt  for Re(s)>-1
public static double Integrate(double start, double end) {
    double currentIntegralValue = 0;
    double dx = 0.0001d; // The size of delta x in the approximation
    double x = start; // A = starting point of integration, B = ending point of integration.

    // Ending conditions for the while loop
    // Condition #1: The value of b - x(i) is less than delta(x).
    // This would throw an out of bounds exception.
    // Condition #2: The value of b - x(i) is greater than 0 (Since you start at A and split the integral 
    // up into "infinitesimally small" chunks up until you reach delta(x)*n.
    while (Math.abs(end - x) >= dx && (end - x) > 0) {
        currentIntegralValue += function(x) * dx; // Use the (Riemann) rectangle sums at xi to compute width * height
        x += dx; // Add these sums together
    }
    return currentIntegralValue;
}

private static double function(double s) {
    double sech = 1 / Math.cosh(s); // Hyperbolic cosecant
    double squared = Math.pow(sech, 2);
    return ((Math.pow(s, 0.5)) * squared);
}

public static double getNewSum(double s){
double constant = Math.pow(2, (2*s)-1) / (((Math.pow(2, s)) -2)*(gamma(1+s)));
    return constant*Integrate(0, 1000);
}

// Gamma Function - Lanczos approximation
public static double gamma(double s){
                double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028,
                                  771.32342877765313, -176.61502916214059, 12.507343278686905,
                                  -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7};
                int g = 7;
                if(s < 0.5) return Math.PI / (Math.sin(Math.PI * s)*gamma(1-s));

                s -= 1;
                double a = p[0];
                double t = s+g+0.5;
                for(int i = 1; i < p.length; i++){
                        a += p[i]/(s+i);
                }

                return Math.sqrt(2*Math.PI)*Math.pow(t, s+0.5)*Math.exp(-t)*a;
        }

//Binomial Co-efficient - NOT CURRENTLY USING
/*
public static double binomial(int n, int k)
{
    if (k>n-k)
        k=n-k;

    long b=1;
    for (int i=1, m=n; i<=k; i++, m--)
        b=b*m/i;
    return b;
} */

// Riemann's Functional Equation
// Tried this initially and utterly failed.
public static double riemannFuncForm(double s) {
double term = Math.pow(2, s)*Math.pow(Math.PI, s-1)*(Math.sin((Math.PI*s)/2))*gamma(1-s);
double nextTerm = Math.pow(2, (1-s))*Math.pow(Math.PI, (1-s)-1)*(Math.sin((Math.PI*(1-s))/2))*gamma(1-(1-s));
double error = Math.abs(term - nextTerm);

if(s == 1.0)
    return 0;
else 
    return Math.pow(2, s)*Math.pow(Math.PI, s-1)*(Math.sin((Math.PI*s)/2))*gamma(1-s)*standardZeta(1-s);
}

}

Answer 1

I think I need to use a different form of the zeta function. When I run the entire program ---

import java.util.Scanner;

public class Test4{

public static void main(String[] args) {
    RiemannZetaMain func = new RiemannZetaMain();
    double s = 0;
    double start, stop, totalTime;
    Scanner scan = new Scanner(System.in);
    System.out.print("Enter the value of s inside the Riemann Zeta Function: ");
    try {
            s = scan.nextDouble();
    } 
    catch (Exception e) {
        System.out.println("You must enter a positive integer greater than 1.");
    }
    start = System.currentTimeMillis();
    if(s >= 2)
        System.out.println("Value for the Zeta Function = " + getStandardSum(s));
    else
        System.out.println("Value for the Zeta Function = " + getRiemannSum(s));
    stop = System.currentTimeMillis();
    totalTime = (double) (stop-start) / 1000.0;
    System.out.println("Total time taken is " + totalTime + " seconds.");
}

// Standard form the the Zeta function.
public static double standardZeta(double s) {
    int n = 1;
    double currentSum = 0;
    double relativeError = 1;
    double error = 0.000001;
    double remainder;

    while (relativeError > error) {
        currentSum = Math.pow(n, -s) + currentSum;
        remainder = 1 / ((s-1)* Math.pow(n, (s-1)));
        relativeError =  remainder / currentSum;
        n++;
    }
    System.out.println("The number of terms summed was " + n + ".");
    return currentSum;
}

public static double getStandardSum(double s){
    return standardZeta(s);
}

// Riemann's Functional Equation
public static double riemannFuncForm(double s, double threshold, double currSteps, int maxSteps) {
double term = Math.pow(2, s)*Math.pow(Math.PI, s-1)*(Math.sin((Math.PI*s)/2))*gamma(1-s);
//double nextTerm = Math.pow(2, (1-s))*Math.pow(Math.PI, (1-s)-1)*(Math.sin((Math.PI*(1-s))/2))*gamma(1-(1-s));
//double error = Math.abs(term - nextTerm);

if(s == 1.0)
    return 0;
else if (s == 0.0)
    return -0.5;
else if (term < threshold) {//The recursion will stop once the term is less than the threshold
    System.out.println("The number of steps is " + currSteps);
    return term;
}
else if (currSteps == maxSteps) {//The recursion will stop if you meet the max steps
    System.out.println("The series did not converge.");
    return term;
}    
else //Otherwise just keep recursing
    return term*riemannFuncForm(1-s, threshold, ++currSteps, maxSteps);
}

public static double getRiemannSum(double s) {
    double threshold = 0.00001;
    double currSteps = 1;
    int maxSteps = 1000;
    return riemannFuncForm(s, threshold, currSteps, maxSteps);
}

// Gamma Function - Lanczos approximation
public static double gamma(double s){
                double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028,
                                  771.32342877765313, -176.61502916214059, 12.507343278686905,
                                  -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7};
                int g = 7;
                if(s < 0.5) return Math.PI / (Math.sin(Math.PI * s)*gamma(1-s));

                s -= 1;
                double a = p[0];
                double t = s+g+0.5;
                for(int i = 1; i < p.length; i++){
                        a += p[i]/(s+i);
                }

                return Math.sqrt(2*Math.PI)*Math.pow(t, s+0.5)*Math.exp(-t)*a;
        }

//Binomial Co-efficient
public static double binomial(int n, int k)
{
    if (k>n-k)
        k=n-k;

    long b=1;
    for (int i=1, m=n; i<=k; i++, m--)
        b=b*m/i;
    return b;
}

}

I notice that plugging in zeta(-1) returns -

Enter the value of s inside the Riemann Zeta Function: -1
The number of steps is 1.0
Value for the Zeta Function = -0.0506605918211689
Total time taken is 0.0 seconds.

I knew that this value was -1/12. I checked some other values with wolfram alpha and observed that --

double term = Math.pow(2, s)*Math.pow(Math.PI, s-1)*(Math.sin((Math.PI*s)/2))*gamma(1-s);

Returns the correct value. It is just that I am multiplying this value every time by zeta(1-s). In the case of Zeta(1/2), this will always multiply the result by 0.99999999.

Enter the value of s inside the Riemann Zeta Function: 0.5
The series did not converge.
Value for the Zeta Function = 0.999999999999889
Total time taken is 0.006 seconds.

I am going to see if I can replace the part for --

    else if (term < threshold) {//The recursion will stop once the term is less than the threshold
    System.out.println("The number of steps is " + currSteps);
    return term;
}

This difference is the error between two terms in the summation. I may not be thinking about this correctly, it is 1:16am right now. Let me see if I can think better tomorrow ....

Answer 2

Ok well we've figured out that for this particular function, since this form of it isn't actually a infinite series, we cannot approximate using recursion. However the infinite sum of the Riemann Zeta series (1\(n^s) where n = 1 to infinity) could be solved through this method.

Additionally this method could be used to find any infinite series' sum, product, or limit.

If you execute the code your currently have, you'll get infinite recursion as 1-(1-s) = s (e.g. 1-s = t, 1-t = s so you'll just switch back and forth between two values of s infinitely).

Below I talk about the sum of series. It appears you are calculating the product of the series instead. The concepts below should work for either.

Besides this, the Riemann Zeta Function is an infinite series. This means that it only has a limit, and will never reach a true sum (in finite time) and so you cannot get an exact answer through recursion.

However, if you introduce a "threshold" factor, you can get an approximation that is as good as you like. The sum will increase/decrease as each term is added. Once the sum stabilizes, you can quit out of recursion and return your approximate sum. "Stabilized" is defined using your threshold factor. Once the sum varies by an amount less than this threshold factor (which you have defined), your sum has stabilized.

A smaller threshold leads to a better approximation, but also longer computation time.

(Note: this method only works if your series converges, if it has a chance of not converging, you might also want to build in a maxSteps variable to cease execution if the series hasn't converged to your satisfaction after maxSteps steps of recursion.)

Here's an example implementation, note that you'll have to play with threshold and maxSteps to determine appropriate values:

/* Riemann's Functional Equation
 * threshold - if two terms differ by less than this absolute amount, return
 * currSteps/maxSteps - if currSteps becomes maxSteps, give up on convergence and return
 * currVal - the current product, used to determine threshold case (start at 1)
 */
public static double riemannFuncForm(double s, double threshold, int currSteps, int maxSteps, double currVal) {
    double nextVal = currVal*(Math.pow(2, s)*Math.pow(Math.PI, s-1)*(Math.sin((Math.PI*s)/2))*gamma(1-s)); //currVal*term
    if( s == 1.0)
        return 0;
    else if ( s == 0.0)
        return -0.5;
    else if (Math.abs(currVal-nextVal) < threshold) //When a term will change the current answer by less than threshold
        return nextVal; //Could also do currVal here (shouldn't matter much as they differ by < threshold)
    else if (currSteps == maxSteps)//When you've taken the max allowed steps
        return nextVal; //You might want to print something here so you know you didn't converge
    else //Otherwise just keep recursing
        return riemannFuncForm(1-s, threshold, ++currSteps, maxSteps, nextVal);
    }
}