Euler Totient Function Research Articles

Monotone Nondecreasing Sequences of the Euler Totient Function

Let M(x) denote the largest cardinality of a subset of {n∈N:n≤x}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{n \\in \\mathbb {N}: n \\le x\\}$$\\end{document} on which the Euler totient functionφ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (n)$$\\end{document} is nondecreasing. We show that M(x)=1+O(loglogx)5logxπ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(x) = \\left( 1+O\\left( \\frac{(\\log \\log x)^5}{\\log x}\\right) \\right) \\pi (x)$$\\end{document} for all x≥10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\ge 10$$\\end{document}, answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function σ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (n)$$\\end{document}.

Algebraicity modulo p of generalized hypergeometric series [formula omitted

Let f(z)=nFn−1(α,β) be the hypergeometric series with parameters α=(α1,…,αn) and β=(β1,…,βn−1,1) in (Q∩(0,1])n, let dα,β be the least common multiple of the denominators of α1,…,αn, β1,…,βn−1 written in lowest form and let p be a prime number such that p does not divide dα,β and f(z)∈Z(p)[[z]]. Recently in [11], it was shown that if for all i,j∈{1,…,n}, αi−βj∉Z then the reduction of f(z) modulo p is algebraic over Fp(z). A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height. In this work, we prove that if p>2dα,β then there is a nonzero polynomial Pp(Y)∈Fp(z)[Y] having degree at most p2nφ(dα,β) and height at most 5n(n+1)!p2nφ(dα,β) such that Pp(f(z)modp)=0, where φ is the Euler's totient function. Furthermore, our method of proof provides us a way to make an explicit construction of the polynomial Pp(Y). We illustrate this construction by applying it to some explicit hypergeometric series.

Construction of Kα-orders including admissible ones on classes of discrete intervals

Ordering relations such as total orders, partial orders and preorders play important roles in a host of applications such as automated decision making, image processing, and pattern recognition. A total order that extends a given partial order is called an admissible order and a preorder that arises by mapping elements of a non-empty set into a poset is called an h-order or reduced order. In practice, one often considers a discrete setting, i.e., admissible orders and h-orders on the class of non-empty, closed subintervals of a finite set Ln={0,1,....,n}. We denote the latter using the symbol In⁎. Admissible orders and h-orders on In⁎ can be generated by the function that maps each interval x=[x_,x‾]∈In⁎ to the convex combination Kα(x)=(1−α)x_+αx‾ of its left and right endpoints. In this paper, we determine a set consisting of a finite number of relevant α's in [0,1] that generate different h-orders on In⁎. For every n∈N, this set allows us to construct the families of all h-orders and admissible orders on In⁎ that are determined by some convex combination. We also provide formulas for the cardinalities of these families in terms of Euler's totient function.

On the equivalence of linear cyclic and constacyclic codes

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Finally, we prove that if gcd⁡(3n,ϕ(3n))=1, where ϕ is the Euler's totient function, then all permutation equivalent constacyclic codes of length n over F4 are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.

The constant of point–line incidence constructions

We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(P,L)≥(c+o(1))|P|2/3|L|2/3, with c≈1.27. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erdős's construction.

Dimensions of popcorn-like pyramid sets

This article concerns the dimension theory of the graphs of a family of functions which include the well-known ‘popcorn function’ and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimensions. As tools in the proofs, we use the Chung—Erdős inequality from probability theory, higher-dimensional Duffin—Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the Hölder distortion between different graphs.

On the mean square average of Dirichlet $L$-function over characters of odd parityin a special case

Evaluating the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of the same parity is an active problem in number theory. Here we explicitly evaluate $\sum_{\chi\text{ odd}}L(3,\chi)$ using certain trigonometric sums and Bernoulli polynomials and express the sum in terms of the Euler totient function $\phi$ and the Jordan totient function $J_s$.

Euler's totient function applied to complete hypergroups

<abstract><p>We study the Euler's totient function (called also the Euler's phi function) in the framework of finite complete hypergroups. These are algebraic hypercompositional structures constructed with the help of groups, and endowed with a multivalued operation, called hyperoperation. On them the Euler's phi function is multiplicative and not injective. In the second part of the article we find a relationship between the subhypergroups of a complete hypergroup and the subgroups of the group involved in the construction of the considered complete hypergroup. As sample application of this connection, we state a formula that relates the Euler's totient function defined on a complete hypergroup to the same function applied to its subhypergroups.</p></abstract>

The average behaviour of a hybrid arithmetic function associated to cusp form coefficients over certain sparse sequence

Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group ? = SL(2,Z), and let ?f (n), ?(n) and ?(n) be the nth normalized Fourier coefficient of the cusp form f , the sum-of-divisors function and the Euler totient function, respectively. In this paper, we investigate the asymptotic behaviour of the following summatory function Sj,b,c(x) := X n=a21 +a22 +a23 +a24 ?x (a1,a2,a3,a4)?Z4 ?j f (n)?b(n)?c(n), where j ? 2 is any given integer. In a similar manner, we also establish other similar results related to normalized coefficients of the symmetric power L-functions associated to holomorphic cusp form f.

Generalizing Lehmer’s totient problem

An important unsolved question in number theory is Lehmer's totient problem that asks whether there exists any composite number n such that \varphi(n)\mid n-1 , where \varphi is the Euler's totient function. It is known that if any such n exists, it must be odd, square-free, greater that 10^{30} , and divisible by at least 15 distinct primes. Such a number must be also a Carmichael number. In this short note, we discuss a group-theoretical analogous problem involving the function that counts the number of automorphisms of a finite group. Another way to generalize Lehmer's totient problem is also proposed.

A Study of the Use of Euler Totient Function in RSA Cryptosystem and the Future of RSA Cryptosystem

At the end of the last century, an innovative two-key cryptosystem called RSA was created due to booming demand for secure remote communication. The core for this encryption system is a mathematical function called the Euler totient function (Euler φ function), introduced by Leonhard Euler. This article will first deduce four theorems related to Euler φ function, and then illustrate the correctness of the RSA cryptosystem using Euler’s φ function. The study then starts with well-known RSA attacks such as the Coppersmith attack, assesses some of RSA’s vulnerabilities, and measures the impact of quantum technologies on RSA. Finally, this study provides projections and recommendations for the future development of RSA based on all evaluation studies completed during the entire research program.

Curious congruences for cyclotomic polynomials

Let \(\Phi _n^{(k)}(x)\) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values \(\Phi _n^{(k)}(1)\) for fixed positive integers n. D. H. Lehmer proved that \(\Phi _n^{(k)}(1)/ \Phi _n(1)\) is a polynomial of the Euler totient function \(\phi (n)\) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that \(\Phi _n^{(k)}(1)/\Phi _n(1)\) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: \(2\Phi ^{(3)}_n(1)\) is divisible by \(\phi (n)-2\). Moreover, if k is greater than 1, then \(\Phi ^{(2k+1)}_n(1)\) is divisible by \(\phi (n)-2k\). The proof depends on a new combinatorial identity for general self-reciprocal polynomials over \({{\mathbb {Z}}}\), which gives rise to a formula that expresses the value \(\Phi _n^{(k)}(1)\) as a \({{\mathbb {Z}}}\)-linear combination of the coefficients in the minimal polynomial of \(2\cos (2\pi /n)-2\). As a supplement, we show the monotonic increasing property of \(\Phi _n(x)\) on \([1,\infty )\) in two ways.

The Euler totient function on Lucas sequences

In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are [Formula: see text], and [Formula: see text]. In 2015, Faye and Luca [Pell numbers whose Euler function is a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] proved that the only Pell numbers whose Euler totient function is another Pell number are [Formula: see text] and [Formula: see text]. Here, we add to these two results and prove that for any fixed natural number [Formula: see text], if we define the sequence [Formula: see text] as [Formula: see text], [Formula: see text], and [Formula: see text] for all [Formula: see text], then the only solution to the Diophantine equation [Formula: see text] is [Formula: see text].

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of T^{-1} as T rightarrow infty . We show that a closed horocycle whose length ell goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate T^{-1} provided that T/ell rightarrow 0 and, for any delta >0, the segment remains larger than max left{ T^{-1/6},left( T/ell right) ^{1/2}right} left( T/ell right) ^{-delta }. We also have an effective result for a smaller range of rates of growth of T and ell . Finally, a number-theoretic identity involving the Euler totient function follows from our technique.

On sums of gcd-sum functions

Let [Formula: see text] denote the greatest common divisor of integers [Formula: see text] and [Formula: see text]. Define [Formula: see text] for any arithmetical function [Formula: see text]. We shall derive several asymptotic expansions of the error terms [Formula: see text] for the partial sums of [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where [Formula: see text] and [Formula: see text] are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function [Formula: see text] and deduce several related results.

ON THE VOLTERRA INTEGRAL EQUATION FOR THE REMAINDER TERM IN THE ASYMPTOTIC FORMULA ON THE ASSOCIATED EULER TOTIENT FUNCTION

This paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the asymptotic formula for the associated Euler totient function into two summands called arithmetic and analytic part respectively.

On the solution of the Volterra integral equation of second type for the error term in an asymptotic formula for arithmetical functions

In 2010, J.Kaczorowski and K.Wiertelak considered the Volterra integral equation of second type for the remainder term in the asymptotic formula for the Euler totient function. The author found that the consideration made by them holds for other remainder terms in the asymptotic formula having certain common properties. In this paper, we first consider the pair of complex-valued arithmetical functions $(a(n),b(n))$ satisfying $b(n) \hspace{-0.1cm} = \sum_{d|n} a(d)n/d$. We prove that the solution of the Volterra integral equation of second type for the error term in the asymptotic formula for $b(n)$ can be obtained when $a(n)$ satisfies some special condition.

An Enhancement to Caesar Cipher using Euler Totient Function

In this modern world every communication including financial transactions are taking place through the open unsecured network. In order to communicate securely through open networks the messages need to be concealed by using the cryptographic methods. The mono alphabetic traditional Cipher is Caesar cipher is the simplest and fairly popular among the available cryptographic algorithms but it is prone to brute force attack and frequency analysis test. In this paper we propose an enhancement to Caesar cipher by using the Euler Totient function to generate the key to encrypt a text file. The Euler Totient function is applied on the total number of lines of the plain text file. The key value is generated from the plain text hence for different plain text the key value is different; this makes it difficult for the adversary to guess the key thus making the encryption scheme robust. It needs to design secured key distribution schemes to be taken up as the extension work of this scheme.

Euler totient function and fermat Euler theorem based an optimized key management scheme for securing mobile agents migration

Euler totient function and fermat Euler theorem based an optimized key management scheme for securing mobile agents migration

Trigonometric Ratios Using Algebraic Methods

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the field. Historically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the first and second kind respectively. So, we briefly present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler's totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discuss these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.