Riemann-Siegle-Zeta

This project computes the non-trivial zeros of the Riemann Zeta function using the Riemann-Siegel formula. The code leverages OpenMP for parallel computation and AVX2 for optimized performance, making it suitable for large-scale computations.

Overview

The Riemann Zeta function, denoted as ζ(s), is one of the most important functions in number theory. Its zeros play a significant role in understanding the distribution of prime numbers. The non-trivial zeros of ζ(s) are located in the critical strip where the real part of s is between 0 and 1. The Riemann Hypothesis conjectures that all non-trivial zeros lie on the critical line, where Re(s) = 1/2.

The Riemann-Siegel formula is used in this project to compute the values of ζ(1/2 + it), for real values of t. This formula is efficient for computing values of ζ(s) at large heights on the critical line.

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

Riemann-Siegel Formula

The Riemann-Siegel formula is an approximation for the zeta function on the critical line. For large values of t, it can be written as:

$$Z(t) \approx 2 \sum_{n=1}^{N} \frac{1}{n^{1/2}} \cos\left( \theta(t) - t \log(n) \right)$$

where theta(t) is the Riemann-Siegel theta function and N is approximately:

$$N = \left\lfloor \sqrt{\frac{t}{2\pi}} \right\rfloor$$

The Riemann-Siegel theta function is defined as:

$$\theta(t) = \arg \left( \Gamma\left( \frac{1}{4} + \frac{it}{2} \right) \right) - \frac{t \log \pi}{2}$$

This formula provides an efficient approximation for large values of t in the computation of

$$\zeta \left( \frac{1}{2} + it \right).$$

https://arxiv.org/pdf/1811.01130
https://arxiv.org/pdf/2201.00342

based on : https://web.viu.ca/pughg/thesis.d/masters.thesis.pdf

@at0m741 and @yroussea