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This Volume is a continuation of the first volume of the 8T.TheGrand Unified Theory of Physics.V2.6Inserts: (Pages 233-234) Certainties Elimination - Breakthrough pg-234.Similar tothe first volume, the second volume is written by the solely by Manor Ohad. The second volume contains three parts. Firstpart - Reflections on the open questions of the 8T. Second Part - Reflections on Quantumphenomena using Number theory.Thirdpart - Reflections on the 8T framework and the Beauty of the finallaws of Nature. The authorincluded a collection ofadditional proofs to the most famous problem in number theory. The counting begins with the second proof as the first already presented in the first volume, Page 96-99 in "Classics". The Purpose of This volume is to Eliminate Certainties.&nbsp

Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number. Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being 36), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the smallest better base being the primorial base six, senary). Quaternary shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, and 3. With this encoding, the complementary digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the base pairs: A↔T and C↔G and can be stored as data in DNA sequence.[2] For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 😊 decimal 9156 or binary 10 00 11 11 00 01 00).

Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show that the Robin inequality is true for all natural numbers $n > 5040$ which are not divisible by any prime between $2$ and $1771559$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality complies that necessarily $q_{m} > e^{41.1015158194}$, $1 < \frac{1.25 \times \log(4.6670857352599751)}{\log q_{m}}+ \frac{\log N_{m}}{\log n}$, $(\log n)^{\beta} < 1.0374812171\times\log(N_{m})$, $\beta \times \frac{\pi^{2}}{8} < \prod_{i=1}^{m} \frac{q_{i}^{2}}{q_{i}^{2} - 1}$ and $n < (4.6670857352599751)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$.

Thispaper present threetheorems on a Lorentz manifoldwhich yield an on pointprediction ofthefine structureconstant, and as the result theawaitedequation of coupling magnitudes.this paper than allow us tounify QM with GR. The equation also predicting the magnitude of any additional element in the series, the next element should stand at ratio of 1/850 compared to the strong interaction. Updated to 7.9.21 &nbsp

Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that \(e,\sin 1\), and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds 2. We construct all \(\aleph _0^{\aleph _0}=\mathfrak {c}\) such series and recover the transcendence of the Davison–Shallit and Cahen constants. Along the way, we mention \(\pi \), the golden ratio, Fermat, Fibonacci, and Liouville numbers, Sylvester’s sequence, Pierce expansions, Mahler’s method, Engel series, and theorems of Lambert, Sierpinski, and Thue-Siegel-Roth. We also make three conjectures.

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a verified proof of the Riemann Hypothesis. These results are supported by the claim that a numerical computer calculation verifies that the subtraction of \[\log (e^{\gamma} \times q_{m} \times r) + e^{\gamma} \times q_{m} \times r \times \log \log (e^{\gamma} \times q_{m} \times r)\] with \[(q_{m} + 1) \times \log (e^{\gamma} \times (r + 1)) + (q_{m} + 1) \times e^{\gamma} \times (r + 1) \times \log \log (e^{\gamma} \times (r + 1))\] is monotonically increasing as much as $q_{m}$ and $r$ become larger just starting with the initial values of $q_{m} = 47$ and $r = 1$, where $q_{m}$ is a prime number and $r$ is a natural number. In this way, we can confirm that the Riemann Hypothesis is true based on computational mathematics using a simple and naive computer assisted proof.

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log n/\log\log\log n$. That is, pr$(n) \sim \log\log n/\log\log\log n$ as $n\to\infty$ on a set of integers of asymptotic density 1. In fact we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer $k$ with $k!\mid \phi(n)$., Comment: 10 pages

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.

We give an extension of a theorem of Cesaro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai's function to number fields.

On the ground of Prime Numbers Classification it is generalized approach to Composite Numbers Factorization presented. The resulting task goes to the different Diophantine equations reducing. Some methods of reducing and its practical application, in particular for Fermat numbers, are demonstrated.

In this study, sequences are used for direct computation of prime numbers. A single equation for generating all prime numbers with the exception of 2 and 3 and composite numbers that are not divisible by 2 or 3 is presented. A periodicity to the indices of the composites generated by that equation is determined. An equation to determine the indices of the composite numbers is derived. The equation for determining the composite indices is then altered to avoid redundancy by observation of its diagonal, when its values are inserted into a matrix that has the indexes (j) for columns and (k) for rows. The two equations are presented in several different forms and are used in conjunction to render a technique for computing prime numbers indefinitely. The validity of the technique is proven up to one billion by computation and the prime number theorem.

If $X$ is a commutative ring with unity, then the unitary Cayley graph of $X$, denoted $G_X$, is defined to be the graph whose vertex set is $X$ and whose edge set is $\{\{a,b\}\colon a-b\in X^\times\}$. When $R$ is a Dedekind domain and $I$ is an ideal of $R$ such that $R/I$ is finite and nontrivial, we refer to $G_{R/I}$ as a \emph{generalized totient graph}. We study generalized totient graphs as generalizations of the graphs $G_{\mathbb{Z}/(n)}$, which have appeared recently in the literature, sometimes under the name \emph{Euler totient Cayley graphs}. We begin by generalizing to Dedekind domains the arithmetic functions known as Schemmel totient functions, and we use one of these generalizations to provide a simple formula, for any positive integer $m$, for the number of cliques of order $m$ in a generalized totient graph. In particular, we prove that the number of cliques of order $m$ in $G_{\mathbb Z/(n)}$ is \[\prod_{k=1}^m\frac{S_{k-1}(n)}{k},\] where $S_r$ is the $r^{\text{th}}$ Schemmel totient function. We then proceed to determine many properties of generalized totient graphs such as their clique numbers, chromatic numbers, chromatic indices, clique domination numbers, and (in many, but not all cases) girths. We also determine the diameter of each component of a generalized totient graph. We correct one erroneous claim about the clique domination numbers of Euler totient Cayley graphs that has appeared in the literature and provide a counterexample to a second claim about the strong domination numbers of these graphs., 16 pages, 0 figures

It is known, that any prime number has presentation in linear and quadratic forms. These properties may be used for finding class subsets of prime numbers. For that it is showed, that all prime numbers simple quadratic forms consist of a2+mb2, m=1,±2,3 . On these grounds it is examination for variants of prime numbers classification. It is discovered eight non-intersecting subsets of prime numbers, in conformity with equivalence classes modulo 24. The proposed classification is used for analyses Mersenne and Fermat numbers and composite numbers.

An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. For the $n$th prime $p_{n}$, the $n$th primorial $p_{n}^{\sharp}$ is defined as the product of the first $n$ primes. In 1999, Odlyzko, Rubinstein and Wolf provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $p_{1}^{\sharp}, p_{2}^{\sharp}, p_{3}^{\sharp}, p_{4}^{\sharp}, p_{5}^{\sharp}, ...$, that is, $2, 6, 30, 210, 2310, ....$ In this paper, we prove that an appropriate form of the Hardy-Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$., 19 pages, 1 table

In this paper, there are 5 sections of tables represented by 5 linear sequence functions. There are two one-variable sequence functions that they are able to represent all prime numbers. The first one helps the last one to produce another three two-variable linear sequence functions. With the help of these three two-variable sequence functions, the last one, one-variable sequence function, is able to set apart all prime numbers from composite numbers. The formula shows that there are infinitely many prime numbers by applying limit to infinity. The three two-variable sequence functions help us to find the factor of all composite numbers.

International audience; For a large class of digital functions f, we estimate the sums Sigma(n

Let $D_{k}$ be a set with $k$ distinct elements of integers such that $d_{1}, Comment: Some small revise

Statistical modeling by asymmetric waves, with variables amplitude and a half-cycle of fluctuation, dynamics of a scatter of block structure of positive part of a number of the integers prime which located in a row of 10 million natural numbers, proved emergence of three stages of growth of the left and right reference points in blocks of binary decomposition of prime numbers. These a reference point settle down on each side from the dividing line in the form of the two in the degree equal to number of the category of a binary numeral system, without unit. The first stage of critical chaos is formed by critical prime numbers 0, 1 and 2. The second stage of an accruing order begins with number 3 and comes to the end with a margin error in 1% at the 1135th category of binary notation for the left reference point. At blocks increasing on length among the integers prime by calculations after the 1135th category there comes the third stage with high definiteness of the beginning and the end of blocks of binary decomposition of positive prime numbers.

In the proof of the correctness of the Riemann hypothesis held strong Godel's incompleteness theorem. In keeping with the ideas of Poja and Hadamard's mathematical inventions, we decided to go beyond the modern achievements of the Gauss law of prime numbers and Riemann transformations in the complex numbers, knowing that at equipotent prime natural numbers will be sufficient mathematical transformations in real numbers. In simple numbers on the top left corner of the incidence matrix blocks are of the frame. When they move, a jump of the prime rate. Capacity of a number of prime numbers can be controlled by a frame, and they will be more reliable digits. In the column 1 i = there is only one non-trivial zero on (0, ) j = ∞ . By the implicit Gaussian "normal"

Fortune’s Conjecture is extended from a relatively short interval after each primorial # P to an infinite numbers of similar intervals on both sides of primorials # n P , where n is a positive integer. In addition, it is shown that for every prime y P in the interval ( ) 2 2 1 1 # , # j j j j n P P n P P + + − + ,there is a number x P in the interval ( ) 2 1 , j j P P + that is also a prime or 1, such that # y j x P n P P = ± . Since n can take infinitely many values, it is highly probable that the reverse of the above theorem is also true. Accordingly, it is conjecture that for every prime j P , there exist a prime x P in the interval ( ) 2 1 , j j P P + that gives a much larger prime when added to or subtracted from the primorial # j P multiplied by an integer n.

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

A method is given for generating families of non-congruent numbers with arbitrarily many prime factors. We then use this method to construct an infinite set of new families of these numbers with prime factors of the form 8 k + 3 .

Objectives: The paper aims to present a survey of both time-honored and contemporary studies on triangular number, factorial, relationship between the two, and some other numbers associated with them. Methods: The research is expository in nature. It focuses on expositions regarding the triangular number, its multiplicative analog – the factorial and other numbers related to them. Findings: Much had been studied about triangular numbers, factorials and other numbers involving sums of triangular numbers or sums of factorials. However, it seems that nobody had explored the properties of the sums of corresponding factorials and triangular numbers. Hence, explorations on these integers, called factoriangular numbers, were conducted. Series of experimental mathematics resulted to the characterization of factoriangular numbers as to its parity, compositeness, number and sum of positive divisors and other minor characteristics. It was also found that every factoriangular number has a runsum representation of length k, the first term of which is (k −1)! + 1 and the last term is (k −1)! + k . The sequence of factoriangular numbers is a recurring sequence and it has a rational closed-form of exponential generating function. These numbers were also characterized as to when a factoriangular number can be expressed as a sum of two triangular numbers and/or as a sum of two squares. Application/ Improvement: The introduction of factoriangular number and expositions on this type of number is a novel contribution to the theory of numbers. Surveys, expositions and explorations on existing studies may continue to be a major undertaking in number theory.

In this paper, we are going to prove a famous problem concerning the prime numbers called Bertrand's postulate. It states that there is always at least one prime, p between n and 2n, means, there exists n < p < 2n where n > 1. It is not a newer theorem to be proven. It was first conjectured by Joseph Bertrand in 1845. He did not find a proof of this problem but made important numerical evidence for the large values of n. Eventually, it was successfully proven by Pafnuty Chebyshev in 1852. That is why it is also called Bertrand-Chebyshev theorem. Though it does not give very strong idea about the prime distribution like Prime Number Theorem (PNT) does, the beauty of Bertrand's postulate lies on its simple yet elegant definition. Historically, Bertrand's postulate is also very important. After Euclid's proof that there are infinite prime numbers, there was no significant development in the prime number distribution. Peter Dirichlet stated the standard form of Prime Number Theorem (PNT) in 1838 but it was merely a conjecture that time and beyond the scope of proof to the then mathematicians. Bertrand's postulate was a simply stated problem but powerful enough, easy to prove and could lead many more strong assumptions about the prime number distribution. Illustrious Indian mathematician, Srinivasa Ramanujan gave a shorter but elegant proof using the concept of Chebyshev functions of prime, υ(x), Ψ(x)and Gamma function, Γ(x) in 1919 which led to the concept of Ramanujan Prime. Later Paul Erdős published another proof using the concept of Primorial function, p# in 1932. The elegance of our proof lies on not using Gamma function yet finding the better approximations of Chebyshev functions of prime. The proof technique is very similar the way Ramanujan proved it but instead of using the Stirling's approximation to the binomial coefficients, we are proving similar results using well-known proving technique the mathematical induction and they lead to somewhat stronger than Ramanujan's approximation of Chebyshev functions of prime. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 85-87

Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken from the discrete or continuous Fourier basis., Comment: 11 pages, 6 figures

This study suggests grouping of numbers that do not divide the number 3 and/or 5 in eight columns. Allocation results obtained from multiplication of numbers is based on column belonging to him. If in the Sieve of Eratosthenes the majority of multiplication of prime numbers result in a results devoid of practical benefit (numbers divisible by 2, 3 and/or 5), in the sieve of prime numbers using algorithms, each multiplication of prime number gives a result in a number not divisible to 2, 3 and/or 5.

We study the representation number for some special sparse graphs. For graphs with a single edge and for complete binary trees we give an exact formula, and for hypercubes we improve the known lower bound. We also study the prime factorization of the representation number of graphs with one edge.

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.

This survey treats two connected questions in analytic number theory: given a set of natural numbers, one may seek numbers with large prime factors in the set. Alternatively, one searches for smooth numbers in the set. Many examples have been studied: the set of values of a polynomial, the set of integers in a short interval, the set of shifted primes p+a and so on. These are discussed at some length, with references to the literature.

The purpose of this work is to give some identities of q -Euler numbers and polynomials. Finally we construct q -zeta functions which interpolate the q -analogue of Frobenius–Euler numbers at negative integers.

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing")., Comment: PDF file generated using XeLaTeX/XeTeX. arXiv admin note: text overlap with arXiv:1409.4222

Let n = p1p2 … pr be a product of r prime numbers which are not necessarily different. We define then an arithmetic function µm(n) bywhere m is a natural number. We further define the function L(s, µm) by the Dirichlet seriesand will show that L(s, µm), (m ≥ 3), has an infinitely many valued analytic continuation into the half plane Re s > ½.

Let ϕ ( n ) \phi (n) denote Euler’s totient function, i.e., the number of positive integers > n >n and prime to n n . We study pairs of positive integers ( a 0 , a 1 ) (a_{0},a_{1}) with a 0 ≤ a 1 a_{0}\le a_{1} such that ϕ ( a 0 ) = ϕ ( a 1 ) = ( a 0 + a 1 ) / k \phi (a_{0})=\phi (a_{1})=(a_{0}+a_{1})/k for some integer k ≥ 1 k\ge 1 . We call these numbers ϕ \phi –amicable pairs with multiplier k k , analogously to Carmichael’s multiply amicable pairs for the σ \sigma –function (which sums all the divisors of n n ). We have computed all the ϕ \phi –amicable pairs with larger member ≤ 10 9 \le 10^{9} and found 812 812 pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other ϕ \phi –amicable pairs can be associated. Among these 812 812 pairs there are 499 499 so-called primitive ϕ \phi –amicable pairs. We present a table of the 58 58 primitive ϕ \phi –amicable pairs for which the larger member does not exceed 10 6 10^{6} . Next, ϕ \phi –amicable pairs with a given prime structure are studied. It is proved that a relatively prime ϕ \phi –amicable pair has at least twelve distinct prime factors and that, with the exception of the pair ( 4 , 6 ) (4,6) , if one member of a ϕ \phi –amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive ϕ \phi –amicable pairs with larger member > 10 9 >10^{9} , the largest pair consisting of two 46-digit numbers.

where c > 1 is not integer, and proved in both cases that there exists k0(c) such that the corresponding problem has solutions if k ≥ k0 and N is sufficiently large. Later Deshouillers [4] and Arhipov and Zhitkov [1] improved Segal’s result on (2). One may also mention the papers of Deshouillers [5] and Gritsenko [7], where the equation (2) in two variables was considered. In 1952 I. I. Piatetski–Shapiro [12] considered (1) with x1, . . . , xk restricted to prime numbers. Let H(c) denote the least k such that the inequality (1) with fixed e > 0 has solutions in prime numbers for every sufficiently large real N . Piatetski–Shapiro proved that

(1997). Are There Functions That Generate Prime Numbers? The College Mathematics Journal: Vol. 28, No. 5, pp. 352-359.

Let consider the Pauli group $\mathcal{P}_q=$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure., Comment: 11 pages

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely$1/q^k$. In a series of recent papers, using the complexity of the multiplicative structure of integers along with a method we developed in 1995 regarding the distribution of subsets of primes in the prime factorization of integers, we initiated new methods allowing for the creation of large families of normal numbers. Here, we further expand on this initiative.

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $\sigma(n) < e^\gamma n\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $\Psi_t,$ a generalization of Dedekind $\Psi$ function defined for any integer $t\ge 2$ by $$\Psi_t(n):=n\prod_{p \vert n}(1+1/p+\cdots+1/p^{t-1}).$$ If $n$ is $t-$free then the sum of divisor function $\sigma(n)$ is $ \le \Psi_t(n).$ We characterize the champions for $x \mapsto \Psi_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\frac{\Psi_t(n)}{n\log\log n}.$ We prove that, for all $t$, there exists an integer $n_1(t),$ such that we have $R_t(N_n)< e^\gamma$ for $n\ge n_1,$ where $N_n=\prod_{k=1}^np_k.$ Further, by combinatorial arguments, this can be extended to $R_t(N)\le e^\gamma$ for all $N\ge N_n,$ such that $n\ge n_1(t).$ This yields Robin inequality for $t=6,\,7.$ For $t$ varying slowly with $N$, we also derive $R_t(N)< e^\gamma.$

The probability that n positive integers, chosen at random, are relatively prime is 1 /ζ("), where ζ is Riemann's function. This is well known when n = 2, but not so well known otherwise. The proofs and associated discussion of the statement allow us to introduce many aspects of elementary number theory. Furthermore, there is an interesting application to Millikan's famous oil‐drop experiment.