This paper concerns the values of the Euler phi-function evaluated simultaneously on k arithmetic progressions a_1 n + b_1, a_2 n + b_2, ..., a_k n + b_k. Assuming the necessary condition that no two of the polynomials a_i x + b_i are constant multiples of each other, we show that there are infinitely many integers n for which phi(a_1 n + b_1) > phi(a_2 n + b_2) > ... > phi(a_k n + b_k). In particular, there exist infinitely many strings of k consecutive integers whose phi-values are arranged from largest to smallest in any prescribed manner. Also, under the necessary condition ad \ne bc, any inequality of the form phi(an+b) < phi(cn+d) infinitely often has k consecutive solutions. In fact, we prove that the sets of solutions to these inequalities have positive lower density.
Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166-181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to E_q(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.
For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.
Let $ϕ(n)$ be the Euler-phi function, define $ϕ_0(n) = n$ and $ϕ_{k+1}(n)=ϕ(ϕ_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $ϕ_k(n)$ is $y$-smooth, conditionally on a weak form of the Elliott-Halberstam conjecture.
The generalized group of units of the ring modulo $n$ was first introduced by El-Kassar and Chehade, written as $U^k(Z_n)$. This allows us to formulate a new generalization to the Euler phi function $φ(n)$, that represents the order of $% U^k(Z_n)$ and it is denoted by $φ^{k}(n).$ In this paper, we introduce this newly defined function, where we compute its explicit form and examine some of its properties similar to that of $φ(n)$. In addition, we study some generalized equations involving $φ^{k}(n)$ where complete solution is given for some equations by considering the general case and others…
A nonempty subset A of {1,2,...,n} is called primitive if gcd(A)=1. Let f(n) and f_k(n) denote, respectively, the number of primitive subsets and the number of primitive subsets of cardinality k of {1,2,...,n}. Recursion formulas and asymptotic estimates are obtained for both functions.
The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[ρ]$ where $ρ$ is the primitive third root of unity $e^{2πi/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[ρ]/(θ)$ for any given Eisenstein integer $θ$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[ρ]/(γ^n)\rig…
We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).
We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite Fourier expansion of P(n) is given in terms of a, n, and the phi(a)-th roots of unity. Properties of the sequence are derived.
The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by \cite{LP} is indeed close to the truth.
Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.
We present an algorithm to invert the Euler function $ϕ(m)$. The algorithm, for a given $n \geq 1$, in polynomial time ``on average'', finds the set $Ψ(n)$ of all solutions $m$ to $ϕ(m) = n$. In fact, in the worst case, $Ψ(n)$ is exponentially large, and cannot be computed in polynomial time. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that there is a polynomial time reduction of the Partition Problem, an NP-complete problem, to the problem of deciding whether $ϕ(m) = n$ has a solution for a small set of integers n. This shows that the problem of de…
An old conjecture of Sierpinski asserts that for every integer k \ge 2, there is a number m for which the equation ϕ(x)=m has exactly k solutions. Here ϕis Euler's totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpinski's conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
In this, paper we obtain a q-analogue of a double inequality involving the Euler gamma function which was first proved geometrically by Alsina and Tomas and then analytically by Sandor
We show that the equation phi(a)=σ(b) has infinitely many solutions, where phi is Euler's totient function and sigma is the sum-of-divisors function. This proves a 50-year old conjecture of Erdos. Moreover, we show that there are infinitely many integers n such that phi(a)=n and sigma(b)=n each have more than n^c solutions, for some c>0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of phi at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distrib…
Translation of the Latin original "Speculationes circa quasdam insignes proprietates numerorum" (1784). E564 in the Enestrom index. In this paper Euler talks about Farey sequences and proves some results about the phi function, the number of positive integers less than and relatively prime to an integer. Euler uses the notation pi instead of phi.
Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …
In this paper we present a simple method for deriving recurrence relations and we apply it to obtain two equations involving the Lerch Phi function and sums of Bernoulli and Euler polynomials. Connections between these results and those obtained in a paper of H.M. Srivastava, M.L. Glasser and V. Adamchik are pointed out, emphasizing the usefulness of this approach with some meaningful examples.
In this didactic note, we describe a procedure to derive successive approximations of $π$ using Euler Beta functions. It is an interesting exercise for undergraduate students, since it involves polynomial roots, integral calculations, inequalities and Euler Beta functions.
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 $p\# \mid ϕ(n)$, where $ϕ$ 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 $k!\mid ϕ(n)$.
In this paper we derive some identities and inequalities on the Möbius mu function. Our main tool is phi functions for intervals of positive integers and their unions.
In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers, including a weak one-sided Tschebyshev inequality. An example shows that there cannot be a prime number theorem in this situation, or even a pair of Tschebyshev inequalities, but it would be very interesting to know if a one-sided Tschebyshev inequality holds.
In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $σ(n)<e^γn\log\log n$ holds for every integer $n>5040$, where $σ(n)$ is the sum of divisors function, and $γ$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\in\cS$. As a special case, we determine the finitely many numbers of the form $n=a^2+b^2$ that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality $n/ϕ(n)<e^γ\log \log n$; since $σ(n…
Heim, Neuhauser, and Tröger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Turán inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.
We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-reg…
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ α\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< β\,\frac{q}{1-q}+\frac{\log(1-q)}…
