Number Theory


Rigorous Proof for Riemann Hypothesis Using Sigma-Power Laws

Authors: John Yuk Ching Ting

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. Recognizing this [complete] relationship amongst all three sets of intercepts enable the simultaneous study on important intrinsic properties and behaviors arising from our derived key formulae coined Sigma-Power Laws in a mathematically consistent manner. This then permit the rigorous proof for Riemann hypothesis to mature as well as allows explanations for x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our 'Virtual container' incorporating the novel Sigma-Power Laws, are some of the important mathematical tools employed in this research paper to successfully achieve our proof. Not least in [again] using the same Virtual container research method in this paper, there are other additional deeply inseparable mathematical connections between the contents of this paper and our cross-referenced 2017-dated publication on the dual source of prime number infiniteness entitled "Solving Polignac's and Twin Prime Conjectures using Information-Complexity Conservation" (

Comments: 23 Pages. This research paper contains the rigorous proof for Riemann hypothesis and explanation for Gram points. It is cross-referenced to "Solving Polignac's and Twin Prime Conjectures using Information-Complexity Conservation" (

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Submission history

[v1] 2017-03-13 03:27:18
[v2] 2017-03-18 01:01:54
[v3] 2017-04-02 00:16:28
[v4] 2017-04-17 06:11:01

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