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.
Highlights
Riemann hypothesis known as the Riemann zeta hypothesis, first formulated by G.F.B Riemann in 1859, is one of the most famous and important but posed a very difficult problem in mathematics despite attracting concentrated effort from many outstanding mathematicians
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
The Riemann hypothesis is concerned with the trivial zeros and states that: The real part of any non-trivial zeros of the Riemann zeta function is 1⁄2 symbolically Re(s) =1/2, lying on the so-called critical line
Summary
Riemann hypothesis known as the Riemann zeta hypothesis, first formulated by G.F.B Riemann in 1859, is one of the most famous and important but posed a very difficult problem in mathematics despite attracting concentrated effort from many outstanding mathematicians. 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. A particular path in a complex plane used to compute the integral It can be represented by the notation ∫(α) f (z)dz which denote the representations, influence of measurements by temporary disturbance during experiments, by the misuse of applied methods, the representation of an infinite process by a finite process just to mention a few. Some other forms of conducted on the Riemann hypothesis to add some asymptotes related to this study are the infinite and clarity to it and investigate the relationship infinitesimal asymptotes.
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