Rigorous Proof for Riemann Hypothesis Using the Novel Sigma-power Laws and Concepts from the Hybrid Method of Integer Sequence Classification

Proposed by Bernhard Riemann in 1859, Riemann hypothesis refers to the famous conjecture explicitly equivalent to the mathematical statement that the critical line in the critical strip of Riemann zeta function is the location for all non-trivial zeros. The Dirichlet eta function is the proxy for Riemann zeta function. We treat and closely analyze both functions as unique mathematical objects looking for key intrinsic properties and behaviors. We discovered our key formula (coined the Sigma-power law) which is based on our key Ratio (coined the Riemann-Dirichlet Ratio). We recognize and propose the Sigma-power laws (in both the Dirichlet and Riemann versions) and the Riemann-Dirichlet Ratio, together with their various underlying mathematically-consistent properties, in providing crucial de novo evidences for the most direct, basic and elementary mathematical proof for Riemann hypothesis. This overall proof is succinctly summarized for the reader by the sequential Theorem I to IV in the second paragraph of Introduction section. Concepts from the Hybrid method of Integer Sequence classification are important mathematical tools employed in this paper. We note the intuitively useful mental picture for the idea of the Hybrid integer sequence metaphorically becoming the non-Hybrid integer sequence with certain criteria obtained using Ratio study.


Introduction
In 1859, the famous German mathematician Bernhard Riemann proposed the Riemann hypothesis.Hilberts problems are a list of 23 mathematical problems published by the German mathematician David Hilbert in 1900.Millennium Prize Problems are 7 problems in mathematics that were stated by the Clay Mathematics Institute in 2000.The unsolved Riemann hypothesis problem belongs to Hilberts eighth problem and is one of the Millennium Prize Problems.The official statement of Riemann hypothesis as a Millennium Prize Problem was given by mathematician Enrico Bombieri.The proof or disproof of Riemann hypothesis would have far-reaching implications, for instance, in number theory especially for the distribution of prime numbers.
Riemann hypothesis refers to the famous conjecture explicitly equivalent to the mathematical statement that the critical line (σ = 1 2 ) in the critical strip (0 < σ < 1) of Riemann zeta function is the location for all of its non-trivial zeros.The critical strip of Riemann zeta (ζ) function can be represented by the one and only one critical line in the middle and an infinite number of other parallel lines on either side of this critical line, with every single line mathematically described by this function with a particular designated sigma (σ) value comprising of real numbers.The contents of this paper provide all necessary evidence for, and are centered on, the following four theorems as the main theme.
Theorem I: The exact same Riemann-Dirichlet Ratio, directly derived from either the Riemann zeta or Dirichlet eta function, is an irrefutably accurate mathematical expression on the de novo criteria for the actual presence [but not the actual locations] of the complete set of (identical) infinite non-trivial zeros in both functions.
Theorem II: Both the near-identical (by proportionality factor-related) Riemann Sigma-power law and Dirichlet Sigmapower law with their derivations based on either the numerator or denominator of Riemann-Dirichlet Ratio have Dimensional analysis (DA) homogeneity only when their common and unknown σ variable has a value of 1 2 as its solution.Theorem III: The σ variable with value of 1 2 derived using the Sigma-power law [from Theorem II above] is the exact same σ variable in Riemann hypothesis which conjectured σ to also have the value of 1 2 (representing the critical line As the Dirichlet eta function is essentially the surrogate for Riemann zeta function, we treat and closely analyze both functions as unique mathematical objects looking for key intrinsic properties and behaviors.We discovered our key formula (coined the Sigma-power law) and our key Ratio (coined the Riemann-Dirichlet Ratio) with the aid of Dimensional analysis, Ratio study, Calculus, and concepts from the novel Hybrid method of Integer Sequence classification.
In so doing we recognize that it is the (i) Sigma-power laws in both the Dirichlet and Riemann versions which are based on either the numerator or denominator part of (ii) Riemann-Dirichlet Ratio, together with their various underlying mathematically-consistent properties, that crucially provide hidden de novo evidences for the most direct, basic and elementary mathematical proof for Riemann hypothesis.
The derivation of the two Sigma(σ)-power laws has a qualitative Dimensional analysis component of equating either the numerator or denominator respective portions from each of the two sub-ratios of the Riemann-Dirichlet Ratio with their underlying complete mathematical expressions.The relevant σ-power laws derived using either the relevant parameter-{2n} numerator or shifted-by-one parameter-{2n-1} denominator is justifiably mathematically equivalent to each other and also being related by a common proportionality constant.Thus this paper could literally be summed up by the one concise sentence "The Dimensional analysis homogeneity property of the Sigma-power law provides the definitive mathematical proof for Riemann hypothesis to be true".
The Riemann zeta function, denoted by ζ(s) with s = σ + ıt, has both its sum input ΣReIm{s}[= Re{s} depicted graphically on the x-axis + Im{s} depicted on the y-axis] and sum output ΣReIm{ζ(s)} [= Re{ζ (s)} depicted on the x-axis + Im{ζ(s)} depicted on the y-axis] constituted by complex numbers.The set of zeros, or roots, in ζ(s) consist of (the easily identifiable) trivial zeros and (the not-so-easily identifiable) non-trivial zeros both of infinite magnitude.
Like any other function, a key point of responsibility is tightly respecting the correct use of Concept of a function and various Representations of that function to validate its special properties.This will enable us to target our main goal of comprehending why all of the non-trivial zeros should lie on a particular vertical straight line called the critical line (σ = 1 2 ).Manifesting as conjugate pairs of non-trivial zeros, they can succinctly be denoted by ζ( 1 2 ± ıt) = 0. Gram points are the other conjugate pairs values on the critical line defined by Im{ζ( 1 2 ± ıt)} = 0 whereby they obey Grams rule and Rossers rule with many other interesting characteristics needing a detailed treatise and constituting a separate topic on its own.In this research article, similar treatment on Gram points is not embarked upon -this will be carried out in our next planned publication after this paper (titled Key role of Dimensional analysis homogeneity in proving Riemann hypothesis and providing explanations on the closely related Gram points) thus establishing continuity and treating these two articles almost as one.In practice, the positive (0 < t < +∞), and numerically equal to the negative (−∞ < t < 0), counterpart of the conjugate pairs for the zeros of ζ(s) and its Gram points is usually quoted, or employed for calculation purposes.
Finally as a bonus, we provide an intuitively useful mental picture for visualizing the idea of the Hybrid integer sequence metaphorically becoming the non-Hybrid (usual "garden-variety") integer sequence when, from Ratio study, the character for each of the numerator / denominator integer sequence in the selected Ratio change from being near-identical Class function [in Hybrid integer sequences, typified by A228186 from The On-line Encyclopedia of Integer Sequences] to identical Class function [in non-Hybrid integer sequences, typified in this paper by A100967 from The On-line Encyclopedia of Integer Sequences, and all infinite series from, or arising out of, the Riemann zeta and Dirichlet eta functions].
Here, the term Class function refers to the format of the integer sequences underlying mathematical expression.We stress from the outset that, for the purpose of this study, at least one mathematically designed Ratio (from Ratio study) having a cyclical nature to it must be present, and that this is a sine-qua-non prerequisite for the particular Ratio to be considered useful or relevant.

Riemann Conjecture and Riemann Hypothesis
In this section, the essential difference between a conjecture and a hypothesis is expounded below.We may risk being seen as pedantic by advocating that the traditionally-dubbed Riemann hypothesis should instead be previously labeled the Riemann conjecture as this entity was chronologically used in the era prior to a rigorous proof being obtained for the conjecture.In other words, once proven only then should a particular conjecture be strictly termed a hypothesis.A colloquial description of Riemann hypothesis and its broad consequences is now typified by us in the following imaginary conversation [adapted from a previous description by Princeton mathematician Peter Sarnak in page 222 of The Riemann hypothesis: The greatest unsolved problem in mathematics (Sabbagh, 2002)]."There must be over five hundred previous papers which start with Assume the Riemann conjecture is true and the conclusion is fantastic.Those conclusions have now become theorems ever since this conjecture has been proven to be true.Riemann conjecture has at long last become Riemann hypothesis.With this one solution we have proven five hundred theorems or more at once."A knowledgeable book planned for the general public entitled Prime Numbers and the Riemann hypothesis (Mazur & Stein, 2016) will have, as the title suggested, a large proportion of its content devoted to various wonderful direct or indirect relationships between prime numbers and the Riemann hypothesis -rather this term should contextually have been stated in the pre-proof era as Riemann conjecture (instead of Riemann hypothesis) with relationships here being its broad consequences or impacts on prime numbers, and vice versa.

Riemann zeta and Dirichlet eta Functions
The infinite series Riemann zeta function ζ(s) [represented by Eqs. ( 1), (2), and (3) below where n = 1, 2, 3,. . ., ∞] is regarded as one of the most influential mathematical objects with great importance in many branches of contemporary science and mathematics.Instead of the commonly or conventionally used z symbol for its complex variable, this variable is denoted traditionally in this paper by s(= σ + ıt) where i = √ − 1 is the imaginary number; σ (consisting of real numbers with values −∞ < σ < +∞) refers to the argument for the real part of s [denoted by Re{s}]; and t (consisting of real numbers with values −∞ < t < +∞) refers to the argument for the imaginary part of s [denoted by Im{s}].
Eq. ( 2) is the Riemann's functional equation satisfying −∞ < σ < ∞ and can be used to find all the trivial zeros on the horizontal line at ıt = 0 and σ = -2, -4, -6, . . ., ∞ [all negative even integer] whereby ζ(s) = 0 because the factor sin( πs 2 ) vanishes.Γ is the gamma function, an extension of the factorial function [a product function denoted by the !notation; n! = n(n-1)(n-2). . .(n-(n-1))] with its argument shifted down by 1, to real and complex numbers.That is, if n is a positive integer, . 3) is defined for all σ > 0 except for a simple pole at proportionality factor, viz.
n s is also known as Dirichlet eta (η) or alternating zeta function.This η(s) function is a holomorphic function of s as defined by analytic continuation and can mathematically be seen to be defined at σ = 1 whereby an analogous trivial zeros [with presence only] for η(s) [and not for ζ(s)] on the vertical straight line σ = 1 are obtained at s = 1 ± i. 2πk log(2) where k = 1, 2, 3, . . ., ∞.In this paper, unless stated otherwise, the symbol 'log' will refer to natural logarithm.As ζ(s) is closely related by the proportionality factor to η(s), it can be seen that all non-trivial zeros of η(s) must be identical to those of ζ(s) -thus the statement that all the non-trivial zeros of η(s) in the critical strip are on the critical line (σ = 1 2 ) is in accordance with [an alternative version of] the Riemann hypothesis is also true.

Figure 1a
Schematically depicted polar graph of ζ( 12 + ıt) in Figure 3a with plot of ζ(s) along the critical line for real values of t running from 0 to 34, horizontal axis: Re{ζ( 1 2 + ıt)}, and vertical axis: There is consistent mathematical symmetry about the horizontal axis and a plot of ζ( 1 2 + ıt) would have revealed an identical mirror image graph reflected on this axis.The first 5 non-trivial zeros in the critical strip are geometrically visualized as the place where the spirals pass through the origin.This phenomenon should occur infinitely often as the real number values for t are also infinite.This has previously been checked for the first 10,000,000,000,000 non-trivial zeros solutions.Computationally this implies, but does not mathematically prove, that the complete set of non-trivial zeros occur at σ = 1 2 .1a as the spirals passing through the origin do not occur anymore when the σ from 0 < σ < 1 is of values other than 1 2 .Riemann hypothesis can also be stated as the condition of total absence of non-trivial zeros in the critical strip when σ satisfy 0 < σ < 1 2 and 1 2 < σ < 1.

Combinatorics Ratio
We initially referred to the proposed role of Combinatorics (in particular, the ubiquitous nature of Permutation with repetition) in Appendix 2 of our medical research paper Supramaximal elevation in B-type natriuretic peptide and its N-terminal fragment levels in anephric patients with heart failure: a case series (Ting & Pussell, 2012).Specifically, this role refers to its place in the mathematically-enriched arguments for the devised 'Blood Volume -B-type natriuretic peptide feedback control system' in providing plausible explanations for the study findings.
The topics of Permutations (P) and Combinations (C) come under Combinatorics.P is an ordered C.There are two types of P and two types of C with their relevant formulae given below, with n = 0, 1, 2,. . ., ∞. Note: (i) The variables n and k here for Combinatorics are chosen to start from 0 -whereas the n variable for the ζ(s) and η(s), and their related functions elsewhere in this paper starts from 1, (ii) The four equations below could be thought of having near-identical 'Class function' properties -belonging to either the P or C class under the Combinatorics umbrella, and apart from Eq. ( 4), all contains the discrete factorial (!) function which we have previously noted above to be intimately related to the continuous Γ function via Γ(n) = (n − 1)!, and (iii) Binomial is C without repetition, often denoted as C k n .
P with repetition: C with repetition: Numerically, Eqs. ( 4) >( 5) >( 6) >( 7) always holds true.Performing Ratio study, we define the selected ratio coined the Combinatorics Ratio = C with repetition C without repetition .An important aspect of Combinatorics Ratio involves its novelty in supplying us with a Hybrid integer sequence (A228186) via the defined inequality relationship "Greatest k >n such that < 2 is a maximum]" as part of the Hybrid method of Integer Sequence classification.This was previously published by us on The On-line Encyclopedia of Integer Sequences website as A228186, https://oeis.org/A228186(Ting, 2013).To the best of our knowledge, A228186 is the first ever Hybrid integer sequence artificially synthesized from the Combinatorics Ratio defined with the inequality criteria.
Hybrid integer sequence A228186 is equal to non-Hybrid integer sequence A100967 except for the 21 'exceptional' terms at positions 0, 11, 13, 19, 21, 28, 30, 37, 39, 45, 50, 51, 52, 55, 57, 62, 66, 70, 73, 77, and 81 with their values given by the relevant A100967 term plus 1. A100967 (Noe, 2004) in The On-line Encyclopedia of Integer Sequences, is defined by Greatest k such that Binomial (2k+1, k-n-1) ≥ Binomial (2k, k) whereby n = 0, 1, 2,. . .,∞, and was authored by Tony Noe.Note (i) the identical 'Class function' [Binomial, which is C without repetition] for numerator and denominator ex- can be seen to be a hybrid of two infinite series A100967(+0) and A100967(+1) -the convention employed here being A100967(+0) and A100967(+1) respectfully symbolizing add 0 and add 1 to every single A100967 term.The totally predictable 21 'exceptional' terms of A228186, possessing deterministic pseudo-randomness and self-similarity properties, makes A228186 a novel true pseudorandom infinite-length integer sequence with highly significant connections to the mathematical fields of Chaos and Fractals.Using logical deduction, the limited number of 21 'exceptional' terms can be explained by the following mathematical constraint -with progressively higher 'exceptional' terms, the fractional part of the Combinatorics Ratio using (k+1) values will overall be monotonously rising steadily approaching a value of 1 (boundary condition), which then limit the total number of possible exceptional terms to just 21.

Riemann-Dirichlet Ratio
Euler formula is commonly stated as e ıx = cos x + ı. sin x.The magnificent Euler identity (where x = π) is e ıπ = cos π + ı. sin π = −1 + 0, commonly stated as e ıπ + 1 = 0.The n s of Riemann zeta function can be expanded to n s = n (σ+ıt) = n σ .et. log(n).ısince n t = e t. log (n) .Apply the Euler formula to n s will result in n s = n σ .(cos(t.log(n)) + ı. sin(t.log(n)) -designated here with the short-hand notation n s (Euler) -whereby n σ is the modulus and t. log(n) is the polar angle.
Apply n s (Euler) to Eq. ( 1), we have n −σ .sin(t.log(n)).As Eq. ( 1) is defined only for σ > 1 where zeros never occur, we will not carry out further treatment related to this subject area.
Apply the trigonometry identity cos(x) − sin(x) = √ 2. sin Eq. ( 10) completely fulfill the 'presence of the complete set of non-trivial zeros' criteria.Rearranging its terms will result in our desired Riemann-Dirichlet Ratio given below.
Denote the left hand side ratio as Ratio R1 (of a 'cyclical' nature) and the right hand side ratio as Ratio R2 (of a 'noncyclical' nature).Then the Riemann-Dirichlet Ratio can be deemed to be representing a more complicated 'dynamic' version of non-Hybrid integer sequence in that besides consisting of identical 'Class function' in each of the two functions when expressed in Ratio R1's numerator and denominator, this first Ratio R1 is again given as an equality to another seemingly different Ratio R2 whose numerator and denominator also consist of identical 'Class function'.One may intuitively think of a Hybrid integer sequence to metaphorically arise from a non-Hybrid integer sequence "in the limit" the non-identical 'Class function' in Hybrid integer sequence becomes the identical 'Class function' in the new non-Hybrid integer sequence.Note the absence and presence of σ variable in Ratio R1 and R2 respectively.
The Riemann-Dirichlet Ratio calculations, valid for all continuous real number values of t, would theoretically result in infinitely many non-Hybrid integer sequences [here arbitrarily] for the 0 <σ<1 critical strip region of interest with n = 1, 2, 3,. . ., ∞ being discrete integer number values, or n being continuous real numbers from 1 to ∞ with Riemann integral applied in the interval from 1 to ∞.This infinitely many integer sequences can geometrically be interpreted to representatively cover the entire plane of the critical strip bounded by σ values of 0 and 1, thus (at least) allowing our proposed proof to be of a 'complete' nature.x-axis = t, y-axis = R1 and R2 values.The R2 lines seem to mathematically constraint the maxima and minima values of R1 but R1 overall sinusoidal landscape is now vastly different to that of Figure 2a -we note here that n goes from 1, 2, 3. . .,∞ and each individual n value could be graphically represented manifesting Fractals with self-similarity.

Sigma-power Law
The γ proportionality factor term in Riemann ζ function, viz. 1 (1−2 1−s ) , can also be expressed with the aid of Euler formula as follows (with the formula for σ = 1 2 substitution depicted last).
The Dirichlet and Riemann σ-power laws are given by the exact formulae in Eqs. ( 18) to ( 21) below with ψ being the same proportionality constant valid for both power laws.We can now dispense with the constant of integration C. Using Dimensional analysis approach we can easily conclude that the 'fundamental dimension' [Variable / Parameter / Number X to the power of Number Y] has to be represented by the particular 'unit of measure' [Variable / Parameter / Number X to the power of Number Y whereby Number Y needs to be of the specific value 1 2 ] for Dimensional analysis homogeneity to occur.This de novo Dimensional analysis homogeneity equates to the location of the complete set of non-trivial zeros and is crucially a fundamental property present in all laws of Physics.The 'unknown' σ variable, now endowed with the value of 1 2 , is treated as Number Y. Dirichlet σ-power law using the {2n} parameter: With the common parameter {2n} cancelling out on both sides, the equation reduces to . sin(t.log(2n) Similarly for the {2n-1} parameter, this equivalent equation is . sin(t.log(2n − 1) Figure 3a Figure 3a.Dirichlet σ-power law: using {2n} parameter, n = 1 situation, x-axis = t, ψ arbitrarily defined with value 1, y-axis = σ-power law values obtained when σ = 0.2 (non-critical line), 1 2 (critical line where all non-trivial zeros lie), and 0.8 (non-critical line) displayed on the usual linear-linear graph.

Figure 3b
Figure 3b.Dirichlet σ-power law: using the same criteria and data to that obtained in Figure 3a except that they are now displayed on the log-linear graph, with log here referring to logarithm in base 10.Note (i) the mathematical requirement of the y-axis in Figure 3b to be linear [and not with log scale] due to the mixture of positive and negative values obtained for y-axis and (ii) the implied mathematical symmetry about the σ = 1 2 value in both Figure 3a and Figure 3b.Finally, the Riemann σ-power law is given by the exact formulae using {2n} and {2n-1} parameters with the γ = (2 1 2 .(cos(t.log(2) + ı. sin(t.log(2))/(2 1 2 .(cos(t.log(2) + ı. sin(t.log(2) − 2) substitution.
We illustrate the Dimensional analysis non-homogeneity property for an σ = 1 4 arbitrarily chosen value [clear-cut case with {2n}-parameter] of Riemann σ-power law lying on a non-critical line (with total absence of non-trivial zeros) in the following formula derived using Eqs.( 17) and ( 21).As Ratio R1 component of Riemann-Dirichlet Ratio is independent of σ variable, unlike the Ratio R2 component of Riemann-Dirichlet Ratio and the γ proportionality factor which are dependent on σ variable, we now note the mixture of 1 4 and 1 2 exponents subtly, but nonetheless, present in this formula indicating Dimensional analysis non-homogeneity.Also the replacement of 1 3 fraction with 2 5 fraction [derived from substituting σ = 1 4 into 1 2(σ+1) ] has occurred.Mathematically, this Dimensional analysis nonhomogeneity property for any real number value of σ, when σ 1 2 and 0 <σ <1, will always be present. [

Conclusions
The seemingly small but utterly essential mathematical step in recognizing and representing a 2-variable function with parameters {2n} or {2n-1} allows crucial moments where cancellation of the relevant "common" parameters in Riemann-Dirichlet Ratio and various Sigma-power laws can occur, further allowing the proper Dimensional analysis process to happen in the absolute correct way.These "common" parameters must be mathematically viewed as (2n) 1 or (2n − 1) 1 , viz. raised to a power (exponent) of 1 which will hamper proper Dimensional analysis if not serendipitously deletedcontrast this scenario with the presence of parameters (2n) 1 2 or (2n − 1) 1 2 , viz. raised to a power (exponent) of 1 2 which will then enable proper Dimensional analysis (homogeneity) to proceed.The mathematical foot-prints (6 steps) of this paper are: Euler product formula: The wider scientific community clearly believes that Riemann hypothesis has numerous and widespread perceived impacts or consequences -we will not elaborate further on this important point.We leave behind a final note by briefly mentioning (i) the beautiful Hadamard product above -the infinite product expansion of ζ(s) based on Weierstrass's factorization theorem, and (ii) the beautiful Euler product formula above connecting Riemann zeta function and prime numbers discovered by Euler -this identity has, by definition, the left hand side being ζ(s) and the infinite product on the right hand side extends over all prime numbers p.The form of the Hadamard product clearly displays the simple pole at s = 1, the trivial zeros at all even negative integers due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ; with the letter γ in the expansion here denoting the Euler-Mascheroni constant -note that this is different to the γ[= 1 (1−2 1−s ) ] role being employed in the main content of this paper above.Note also that with the second simpler infinite product expansion formula of Hadamard, to ensure convergence, the product should be taken over "matching pairs" of zeroes, i.e. the factors for a pair of zeroes of the form ρ and 1 − ρ should be combined.

Figure 2a Figure
Figure 2a Figure 2a.Riemann-Dirichlet Ratio displayed as Ratio R1 and Ratio R2 for σ = 1 2 at n = 1 situation.x-axis = t, y-axis = R1 and R2 values.Note R2 lines seem to mathematically constraint the maxima and minima values of R1.

Figure 2b Figure
Figure 2bFigure2b.Riemann-Dirichlet Ratio displayed as Ratio R1 and Ratio R2 for σ = 1 2 at n = 2 situation.x-axis = t, y-axis = R1 and R2 values.The R2 lines seem to mathematically constraint the maxima and minima values of R1 but R1 overall sinusoidal landscape is now vastly different to that of Figure2a-we note here that n goes from 1, 2, 3. . .,∞ and each individual n value could be graphically represented manifesting Fractals with self-similarity.

Step 1 :
Riemann zeta or Dirichlet eta function [for the critical strip 0 < σ < 1] → Step 2: Riemann zeta or Dirichlet eta function [with Euler formula application] → Step 3: Riemann zeta or Dirichlet eta function [simplified and identical version specifically indicating the criteria for the presence of the complete set of non-trivial zeros] → Step 4: Riemann-Dirichlet Ratio [in discrete summation format] → Step 5: Riemann-Dirichlet Ratio [in continuous integral format] → Step 6: Riemann Sigma-power law and Dirichlet Sigma-power law [both with Dimensional analysis homogeneity].correct solution to Riemann zeta and Dirichlet eta function as simplified and identical versions specifically indicating the criteria for the presence of the complete set of non-trivial zeros [in Step 3 of our mathematical foot-prints].