Euler Totient Function Research Articles

An Abstraction of Whitney's Broken Circuit Theorem

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Mobius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.

What are the last digits of …?

We propose a class assignment where students are asked to construct and implement an efficient algorithm to calculate the last digits of a positive integral power of a positive integer. The mathematical prerequisites for this assignment are very limited: knowledge of remainder calculus and the binary representation of a positive integer. The periodicity of the last digits is studied by means of the Euler totient function and the Carmichael function.

A GROUP SUM INEQUALITY AND ITS APPLICATION TO POWER GRAPHS

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite group of order $n$, and let $\text {C}_n$ be the cyclic group of order $n$. For $g\in G$, let ${\mathrm{o}}(g)$ denote the order of $g$. Let $\phi $ denote the Euler totient function. We show that $\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$, with equality if and only if $G$ is isomorphic to $\text {C}_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of bidirectional edges in its directed power graph.

On additive combinatorics of permutations of ℤ<sub>n</sub>

Special issue PRIMA 2013 Let ℤ<sub>n</sub> denote the ring of integers modulo n. A permutation of ℤ<sub>n</sub> is a sequence of n distinct elements of ℤ<sub>n</sub>. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ<sub>n</sub>, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2<sup>k</sup>≤s(n)≤n!· 2<sup>-(n-1)/2</sup>((n-1)/2)! and 2<sup>(n-1)/2</sup>·(n-1 / 2)!≤t(n)≤ 2<sup>k</sup>·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.

Symmetrical parallel RC Multiple-T Frequency Combinations

For a given number of resistance alms n, and capacitance arms nc in a symmetrical parallel RC multiple-T, it is possible to know the maximum number of dissimilar resonant frequency combinations theoretically from a formula derived using a combinational technique. Verification of the results is done by counting the number of dissimilar frequencies for the ratio nr/ nc. WM=WD (nrnc)1/2 and also from frequencies practically determined. The technique common to both these methods is the combinatorial nne as used in deriving Euler's Totient function in the theory of numbers.

ON SUMMATION OF SUBSERIES OF THE RIEMANN ZETA FUNCTION

Three classes of subseries of the Riemann zeta function are evalu- ated in closed form. All the results are expressed in terms of the Riemann zeta function itself and powers of �. By using the Euler totient function, we prove that these three classes are unique subseries that exhibit this kind of closed form.

Numerical Tribute to Achievements of Euler

This paper aims to make a tribute to one of the world's brightest personalities, the mathematical physicist Leonhard Euler (1707-1783). Our purpose is to expose, in a concise and entertaining way by utilizing modern computational techniques, a better understanding of the influence of Euler; also, to establish the remarkable significance in the history of science. A first analysis was done with the series that defines Euler and Bernoulli numbers and polynomials of Bernoulli and Euler; an additional result is the characterization of the functions that lead to the Euler-Mascheroni constant. In hydrodynamics it is also feasible to evaluate graphically the connection between sizes in diameter and the exit angle of the height of Euler for turbo-machines. In differential equations of Cauchy-Euler solutions in the cases of distinct real roots and complex roots are generated. Furthermore we report the generation of the Fourier series and the Fourier transform calculated by using direct Matlab commands. In calculus of variations, it is conceivable to obtain plots from a problem of the Euler Lagrange equations. Finally, Euler totient function is graphically analyzed. We seek the following benefits: The association between Euler numbers and generating functions is more effectively understood, also the nature of the Euler constant and the Euler function The Fourier transform is more comprehensible; also, it can be established that Matlab is an alternative way to study the conceptual development of achievements of physics and mathematics. We show that the abundance of Euler work and the history of mathematics, acquire better knowledge with the application of numerical tools

Aesthetics and quality of numbers using the primety measure

The Farey sequences can be used to create the Euler totient function, by identifying the fractions for number n that did not occur in all Farey sequences up to n - 1. This function creates, when divided by n - 1, what is here called the primety measure, which is a measure of how close to being a prime number n is. The quality of a number sequence can be determined using the primety, which is further generalised to real numbers through the use of real numbered Farey sequences. Ambiguous and interesting results are obtained in this process; for instance it seems that all integers approach full primety when investigated in the real domain. The corresponding numerical sequences are furthermore shown to have interesting artistic properties. In particular the fractal behaviour is interesting. To further investigate fractal ambiguity, common fractals are extended with random elements, which render softer or more varied shapes.

GENERALIZED CULLEN NUMBERS WITH THE LEHMER PROPERTY

We say a positive integer n satisfies the Lehmer property if <TEX>${\phi}(n)$</TEX> divides n - 1, where <TEX>${\phi}(n)$</TEX> is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form <TEX>$D_{p,n}=np^n+1$</TEX>, for a prime p and a positive integer n, or of the form <TEX>${\alpha}2^{\beta}+1$</TEX> for <TEX>${\alpha}{\leq}{\beta}$</TEX> does not satisfy the Lehmer property.

UNSOLVED PROBLEMS

ABSTRACT In this paper there are given problems from the Unsolved Problems Section on the homepage of the journal Uniform Distribution Theory &lt;http://www.boku.ac.at/MATH/udt/unsolvedproblems.pdf&gt; It contains 38 items and 5 overviews collected by the author and by Editors of UDT. They are focused at uniform distribution theory, more accurate, distri- bution functions of sequences, logarithm of primes, Euler totient function, van der Corput sequence, ratio sequences, set of integers of positive density, exponen- tial sequences, moment problems, Benford’s law, Gauss-Kuzmin theorem, Duffin- Schaeffer conjecture, extremes fQ fQ F(x,y)dg(x,y) over copulas g(x,y), sum- -of-digits sequence, etc. Some of them inspired new research activities. The aim of this printed version is publicity.

Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.

Normal coverings and pairwise generation of finite alternating and symmetric groups

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(n) or an alternating group Alt(n), γ(G)⩾cn. This improves results of the first two authors who had earlier proved that aφ(n)⩽γ(G)⩽2n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(G) of a set X of conjugacy classes of G=Sym(n) or Alt(n) such that any pair of elements from distinct classes in X generates G, namely cn⩽κ(G)⩽2n/3.

Clique Dominating Sets of Euler Totient Cayley Graphs

Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Using the number theoretic function Euler totient function we have defined an Euler totient Cayley graph and in this paper we study the Clique domination parameters of Euler totient Cayley graphs.

On a generalization of the Euler totient function

For a general polynomial Euler product F(s) we define the associated Euler totient function φ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of φ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).

Composed products and factors of cyclotomic polynomials over finite fields

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

SummaryWe introduce a generalization of Euler's totient function that, when applied to an integer n ≥ 2, can be written as a polynomial in the prime factors of n. We then show how cyclotomic and complete homogeneous symmetric polynomials appear as factors of these polynomials when n has at most two distinct prime divisors.

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

A Probabilistic Look at Series Involving Euler's Totient Function

Abstract.We use a probabilistic method to evaluate the limit of

Substitutions with Vanishing Rotationally Invariant First Cohomology

The cohomology groups of tiling spaces with three‐fold and nine‐fold symmetries are obtained. The substitution tilings are characterized by the fact that they have vanishing first cohomology group in the space of tilings modulo a rotation. The rank of the rational first cohomology, in the tiling space formed by the closure of a translational orbit, equals the Euler totient function evaluated at N if the underlying rotation group is ZN. When the symmetries are of crystallographic type, the cohomologies are infinitely generated.