Melvyn B. Nathanson
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.
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This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.
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…
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.
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 …
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…
We examine the sums $S(k,\,n)$ of the $k-$th powers of the $ϕ(n)$ integers $α_1<α_2<\cdots<α_{ϕ(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.
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).
For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-γ} ϕ(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $γ$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $ϕ(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold.
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.
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.
In this work we discuss Dirichlet products evaluated at Fibonacci numbers. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's totient function, an upper bound on the number of primitive prime divisors and representations of some related Euler products. Moreover, we sum functions over all primitive divisors of a Fibonacci number and obtain a non--trivial fixed point of this operation.
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…
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.
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.
Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.
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…
In this paper, we study the value-distributions of $L$-functions of holomorphic primitive cusp forms in the level aspect. We associate such automorphic $L$-functions with probabilistic models called the random Euler products. First, we prove the existence of probability density functions attached to the random Euler products. Then various mean values of automorphic $L$-functions are expressed as integrals involving the density functions. Moreover, we estimate the discrepancies between the distributions of values of automorphic $L$-functions and those of the random Euler products.
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 …
Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{π^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $Ω_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $ϕ(n)$.
For Dirichlet $L$-functions in $\mathbb{F}_q [T]$ we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the $2k$-th moment of the Dirichlet $L$-functions at $\frac{1}{2}$, averaged over primitive characters of modulus $R$, is asymptotic to (as $\mathrm{deg} R \longrightarrow \infty$) the $2k$-th moment of the Euler product multiplied by the $2k$-th moment of the Hadamard product. We explicitly obtain the main term of the $2k$-th moment of the Euler product, and we conjecture via random matrix theory the main term of the $2k$-th moment of the Hadamard pr…
Euler constants from primes in arithmetic progression
本文系统研究由算术级数中素数定义的生成级数的欧拉常数γ(d,a)，给出其与一般狄利克雷级数欧拉常数的关系，并进行数值计算与实例讨论。
The distribution of $\gcd(n,ϕ(n))$
本文研究 $\gcd(n,\phi(n))$ 及其因子的分布，推广了 Erdős 和 Pollack 的先前结果。
Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_F of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H^n(U,Fp) as an Fp[G_F/U]-module for all n in N. Previously this structure was known only for n=1, and until recently the structure even of H^1(U,Fp) was determined only for F a local field, a case settled by Borevic and Faddeev in the 1960s. We apply these results to study partial Euler-Poincare characteristics of open subgroups N of the maximal pro-p quotient T of G_…
Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of …
Let $L(s,χ)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $χ$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,χ) \vert$ using the values of Euler's $Γ$ function. We also introduce an alternative way of computing $\log Γ(x)$ and $ψ(x)= Γ^\prime/Γ(x)$,$x\in(0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $\vert L(1,χ) \vert$, where $χ$ runs over the non trivial primitive Dirichlet characters $\bmod\ q$, f…
On a relation to the Riemann Hypothesis and an analytic part for the divisor function
研究sigma_1(n)求和函数相关的解析部分，并在黎曼猜想下给出显式上界。
It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^\pm_{1,1}\Zbb[i]. The resulting billiard map \tilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We…
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the prime factors of the argument, the explicit form of which is proven in this paper. Subsequently it is shown how the procedure can be generalized to the discrete Fourier transform of a function of the greatest common divisor. From this representation some interesting relations concerning the Euler totient function and generalizations thereof are established.
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