Another Casual Coder

Riemann's Zeta Function is at the center of the Riemann's Hypothesis which says that all the zeroes for the Zeta Function are of the form ki + 1/2 (in other words, the real part of all zeroes for the Zeta Function is 1/2). The importance of this hypothesis is that, if confirmed to be true (which most mathematicians believe it to be true), it can provide a very well-defined structure for the constructions of Prime Numbers. For example, one of the corollaries of proving the Riemann hypothesis is that we'll have a precise way to determine the number of primes less than a given M. Hence one can then use this formula to calculate any number of primes between two given numbers N and M. The mathematical importance of finding a well-structured behavior for prime numbers is monumental: prime numbers are the basis of number theory, and despite them being infinite (proved first by Euclid in 300 BCE) and seeming random, it is believed that they follow a somewhat (yet to be understood) well defined structure in nature.

The Zeta Function is an infinite summation consisting of the following formula:

Z(s) = 1^-s + 2^-s + 3^-s + ...

It is defined over the set of complex numbers. Plotting the zeta function over the set of real numbers is an easy task but somewhat not a very interesting curve. Basically one way is to bound the zeta function to a certain M (won't make much of a difference since the tail elements of the Zeta function tend to zero very quickly). As you can see below, it comes down very quickly from an +infinite value of Y, approaching a limit of 1. Code is down below, cheers, ACC.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace BoundedZetaFunction
{
 internal class Program
 {
 static void Main(string[] args)
 {
 for (double s = -20.0; s <= 20.0; s += 0.1) { double val = BoundedZeta(s, 30); Console.WriteLine("{0};{1}", s, val); } } static double BoundedZeta(double s, int n) { double retVal = 0; for (int i = 1; i <= n; i++) retVal += Math.Pow(i, -s); return retVal; } } }