Blog Entry Monday, July 15, 2024, (c) James Pate Williams, Jr. Some Values of the Riemann Zeta Function Computed with Fast and Very Slow Algorithms

// RiemannZetaFunctionWin32Console.c (c) Saturday, July 13-15, 2024
// James Pate Williams, Jr. some computations use 999,999,998 terms

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef signed long long sll;

// https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function#cite_note-7

long double EvenZeta(int n)
{
    sll A[12] = { 0ll,
        6ll, 90ll, 945ll, 9450ll, 93555ll, 638512875ll,
        18243225ll, 325641566250ll, 38979295480125ll,
        1531329465290625ll, 13447856940643125ll };
    sll B[12] = { 0ll, 1ll, 1ll, 1ll, 1ll, 1ll, 691ll, 2ll,
        3617ll, 43867ll, 174611ll, 155366ll };
    long double pi = 4.0 * atan(1.0);

    return (long double)B[n / 2] * powl(pi, n) / (long double)A[n / 2];
}

long double Zeta(long s, long *terms)
{
    int n = 0;
    long double sum = 0.0;

    *terms = 0;

    for (n = 1000000000; n >= 2; n--)
    {
        sum += 1.0 / powl(n, s);
        (*terms)++;
    }

    return sum + 1.0;
}

void PrintEvenZetaValue(int n)
{
    clock_t time0 = clock();
    long double ez = EvenZeta(n);
    clock_t time1 = clock();
    double runtime = ((double)time1 - time0) / CLOCKS_PER_SEC;
    printf_s("Runtime in seconds to compute Zeta(%ld) = %16.15Lf: %Lf\n",
        n, ez, runtime);
}

void PrintZetaValue(int n)
{
    clock_t time0 = clock();
    long terms = 0;
    long double zeta = Zeta(n, &terms);
    clock_t time1 = clock();
    double runtime = ((double)time1 - time0) / CLOCKS_PER_SEC;
    printf_s("Runtime in seconds to compute Zeta(%ld) = %16.15Lf terms = %ld: time = %Lf\n",
        n, zeta, terms, runtime);
}

int main()
{
    PrintEvenZetaValue(2);
    PrintEvenZetaValue(4);
    PrintEvenZetaValue(6);
    PrintEvenZetaValue(8);
    PrintZetaValue(2);
    PrintZetaValue(4);
    PrintZetaValue(6);
    PrintZetaValue(8);
    PrintZetaValue(3);
    PrintZetaValue(5);
    PrintZetaValue(7);
    PrintZetaValue(9);
    return 0;
}