Number Theory

    

On the Validity of the Riemann Hypothesis.

Authors: Khalid Ibrahim

In this paper, we have established a connection between The Dirichlet series with the Mobius function $M (s) = \sum_{n=1}^{\infty} \mu (n) /n^s$ and a functional representation of the zeta function $\zeta (s)$ in terms of its partial Euler product. For this purpose, the Dirichlet series $M (s) $ has been modified and represented in terms of the partial Euler product by progressively eliminating the numbers that first have a prime factor 2, then 3, then 5, ..up to the prime number $p_r $ to obtain the series $M(s,p_r)$. It is shown that the series $M(s)$ and the new series $M(s,p_r)$ have the same region of convergence for every $p_r$. Unlike the partial sum of $M(s)$ that has irregular behavior, the partial sum of the new series exhibits regular behavior as $p_r$ approaches infinity. This has allowed the use of integration methods to compute the partial sum of the new series and to examine the validity of the Riemann Hypothesis.

Comments: 56 Pages.

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Submission history

[v1] 2013-10-15 21:26:42
[v2] 2013-10-17 00:02:55
[v3] 2013-10-27 21:16:53
[v4] 2013-11-11 00:53:52
[v5] 2013-11-20 22:55:18
[v6] 2014-02-02 19:23:32
[v7] 2014-02-08 05:38:17
[v8] 2014-03-08 23:06:44
[v9] 2014-03-16 23:16:18
[vA] 2014-03-29 04:08:43
[vB] 2014-05-11 01:18:52
[vC] 2014-06-06 23:30:08
[vD] 2014-07-12 02:02:14
[vE] 2014-07-22 23:36:21
[vF] 2014-07-27 12:30:30
[vG] 2014-10-02 00:30:39
[vH] 2014-10-19 05:13:37
[vI] 2014-11-16 05:04:38
[vJ] 2018-04-21 23:47:55
[vK] 2018-06-08 04:00:24
[vL] 2018-07-12 05:01:01 (removed)
[vM] 2018-07-18 05:59:53

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