A Numerical Test on the Riemann Hypothesis with Applications
N. K. Oladejo and I. A. Adetunde
DOI : 10.3844/jmssp.2009.47.53
Journal of Mathematics and Statistics
Volume 5, 2009
Problem statement: The Riemann hypothesis involves two products of the zeta function ζ(s) which are: Prime numbers and the zeros of the zeta function ζ(s). It states that the zeros of a certain complex-valued function ζ (s) of a complex number s ≠ 1 all have a special form, which may be trivial or non trivial. Zeros at the negative even integers (i.e., at S = -2, S = -4, S = -6...) are called the non-trivial zeros. The Riemann hypothesis is however concerned with the trivial zeros. Approach: This study tested the hypothesis numerically and established its relationship with prime numbers. Results: Test of the hypotheses was carried out via relative error and test for convergence through ratio integral test was proved to ascertain the results. Conclusion: The result obtained in the above findings and computations supports the fact that the Riemann hypothesis is true, as it assumed a smaller error as possible as x approaches infinity and that the distribution of primes was closely related to the Riemann hypothesis as was tested numerically and the Riemann hypothesis had a positive relationship with prime numbers.
© 2009 N. K. Oladejo and I. A. Adetunde. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.