A Theory to Refute the Riemann Hypothesis

A Theory to Refute the Riemann Hypothesis Stylianos Touloumidis [e-mail] October 23, 2024 Abstract This paper introduces a new theory regarding the distribution of the zeros of the Riemann zeta function on the critical line, challenging the current assumptions about the structure of the zeta function. The classical approach, which incorporates both prime and non-prime numbers into the calculations, obscures the actual zeros and results in computations that approach but never reach the true zeros. By isolating the prime numbers, it becomes apparent that the imaginary part of the zeros on the critical line is directly correlated with the primes. The higher the prime, the greater the distance between the zeros. This discovery reveals a pattern that contradicts previous assumptions about the distribution of the zeros in the zeta function. At the core of this investigation is the definition of the Riemann zeta function, which is defined for complex numbers as follows: \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} This function is deeply connected to prime numbers, as expressed in Euler’s product formula: \zeta(s) = \prod_{p \, \text{prime}} \left( 1 - \frac{1}{p^s} \right)^{-1} This work demonstrates that the distribution of primes entirely determines the zeros on the critical line. The spacing between the zeros corresponds to the differences between prime numbers, and this gap increases indefinitely, potentially refuting the Riemann Hypothesis.