Research Article Open Access

A Numerical Test on the Riemann Hypothesis with Applications

N. K. Oladejo and I. A. Adetunde

Abstract

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.

Journal of Mathematics and Statistics
Volume 5 No. 1, 2009, 47-53

DOI: https://doi.org/10.3844/jmssp.2009.47.53

Submitted On: 25 October 2008 Published On: 31 March 2009

How to Cite: Oladejo, N. K. & Adetunde, I. A. (2009). A Numerical Test on the Riemann Hypothesis with Applications. Journal of Mathematics and Statistics, 5(1), 47-53. https://doi.org/10.3844/jmssp.2009.47.53

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Keywords

  • Riemann hypothesis
  • zeta function
  • gamma function
  • errors
  • prime
  • asymptote
  • integral