Zeros and Primes: An Isolated Examination to Refute the Riemann Hypothesis

Analysis of the Riemann Zeta Function and the Critical Zeros: A New Isolated Approach

This paper discusses the analysis of the Riemann zeta function and the zeros that occur along the critical line . The Riemann Hypothesis, formulated by Bernhard Riemann in 1859, remains one of the most profound unanswered questions in mathematics. The hypothesis asserts that all non-trivial zeros of the zeta function possess a real part of . These zeros play a crucial role in the distribution of prime numbers.

In previous mathematical research, zeros have often been considered globally, without fully accounting for the specific significance of primes as isolated computational units. This paper addresses why previous computational methods have not yielded the desired results and presents a new method for isolating prime numbers, leading to an exact computation of the zeros.

The classical computational approach for determining the zeros of the Riemann zeta function treated all natural numbers continuously, neglecting the gaps between prime numbers. These gaps resulted in distortions in the calculations, as the structure of primes and their role in the distribution of zeros were not analyzed in isolation. It has been overlooked that each prime must be considered individually to accurately determine the zeros. The continuous approach often yielded calculations near zero but failed to correctly represent the critical zeros.

The method presented here introduces a new approach by considering each prime in isolation. The zeros of the zeta function are individually computed along the critical line for each prime number. This method demonstrates that the zeros for each prime continue regularly and grow to infinity. This refutes the previous notion that zeros can be viewed globally and highlights that the isolated computation of primes provides an exact pathway to determining the zeros.

The main formula used for the computation of the zeros is given by:

s_n = 98.0 + n \cdot \text{prime}

The computation of the zeros was performed in two steps:

1. Symbolic computations were conducted to analyze the Riemann zeta function for complex arguments. The symbolic zeta function for complex numbers is defined by the formula:

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

2. In a second step, the zeta function was numerically computed for the first 10 prime numbers. It became evident that the calculation of zeros must be conducted in isolation for each prime to yield accurate results.