Smooth values of the iterates of the Euler's Phi function

On the iterates of shifted Euler's function

Let $φ$ be the Euler's function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=φ(x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for every initial value $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=φ(x_{n+1})+φ(x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that $k$ is even.

Primitive sets and an Euler phi function for subsets of {1,2,...,n}

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.

An Euler phi function for the Eisenstein integers and some applications

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…

On common values of phi(n) and sigma(n), I

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).

$ϕ$ 和 $σ$ 迭代的原像数量

The number of preimages of iterates of $ϕ$ and $σ$

本文将 Euler 函数 $ϕ$ 和 $σ$ 原像数量的渐近上界扩展至其迭代情形，并借此证明了 Konick 和 Katai 论文中的相关猜想及推论。

Galois representations from pre-image trees: an arboreal survey

Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges according to the action of phi. The absolute Galois group of K acts on T by tree automorphisms, giving a subgroup G(phi) of the group Aut(T) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G(p…

The average number of divisors of the Euler function

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.

Common values of the arithmetic functions phi and sigma

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…

Transcendence of values of the iterated exponential function at algebraic points

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \] converges to an algebraic number is approximated by $(e-1/e)φ(k)$. Here $φ(k)$ denotes Euler's totient function.

On common values of $ϕ(n)$ and $σ(m)$, II

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.

Complexity of Inverting the Euler Function

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…

Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series

In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …

Euler-Apéry型多重zeta星值与多重$t$-星值的显式求值

Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values

本文通过迭代积分建立递推关系，进而给出Euler-Apéry型多重zeta星值及其参数变体和多重$t$-星值的显式求值。

Eventually stable rational functions

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and alpha any point not periodic under phi (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the con…

Speculations on some characteristic properties of numbers

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.

Radically weakening the Lehmer and Carmichael conditions

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 …

Recurrence relations for the Lerch Phi function and applications

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.

Galois theory of quadratic rational functions

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in t…

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_m$, we define two new functions, denoted $R_m$ and $H_m$, that arise from iterating $L_m$. Roughly speaking, $R_m$ counts the number of iterations of $L_m$ needed to reach either $0$ or $1$, and $H_m$ takes the value (either $0$ or $1$) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer $m$, the function $H_m$ is completely multiplicative. We then int…

欧拉常数的多对数类似物

Polylogarithmic Analogues of Euler's Constant

引入一族常数 C_m，作为欧拉常数的多对数类似物，研究其性质并建立与伽马函数相关的迭代对数积分表示，进一步引入多对数zeta势和多对数伽马函数，并应用于将C_m与涉及黎曼zeta函数的狄利克雷级数的特殊值联系起来。

$\gcd(n,\phi(n))$ 的分布

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。

Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analog…

Computing the inverses, their power sums, and extrema for Euler's totient and other multiplicative functions

We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power sums (e.g., cardinality) or extrema, without direct enumeration of the inverses. We illustrate our algorithm with Euler's totient function $φ(\cdot)$ and the $k$-th power sum of divisors $σ_k(\cdot)$. For example, we can establish that the number of solutions to $σ_1(x) = 10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, while it is …

关于黎曼猜想的一个关系以及除数函数的解析部分

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。

Self-intersection of the relative dualizing sheaf on modular curves X(N)

Let $N\geq 3$ be a composite, odd, and square-free integer and let $Γ$ be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular curve of genus $g_Γ$ associated to $Γ$. In this article, we study the Arakelov invariant $e(Γ)=\barω^2/φ(N)$, with $\barω^2$ denoting the self-intersection of the relative dualizing sheaf for the minimal regular model of $X(N)$, equipped with the Arakelov metric, and $φ(N)$ is the Euler's phi function. Our main result is the asymptotics $e(Γ) = 2g_Γ\log(N) + o(g_Γ\log(N))$, as the level $N$ tends to infinity.

New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series

We consider relations between the pairs of sequences, $(f, g_f)$, generated by the Lambert series expansions, $L_f(q) = \sum_{n \geq 1} f(n) q^n / (1-q^n)$, in $q$. In particular, we prove new forms of recurrence relations and matrix equations defining these sequences for all $n \in \mathbb{Z}^{+}$. The key ingredient to the proof of these results is given by the statement of Euler's pentagonal number theorem expanding the series for the infinite $q$-Pochhammer product, $(q; q)_{\infty}$, and for the first $n$ terms of the partial products, $(q; q)_n$, forming the denominators of the rational …

On the periodicity of a class of arithmetic functions associated with multiplicative functions

Let $k\ge 1,a\ge 1,b\ge 0$ and $ c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $c {\rm lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,φ}$, and determine the smallest period of $g_{k,φ}$, where $φ$ is the Euler phi function.

Multiple Dedekind Zeta Functions

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are int…

Extremal discrepancy behavior of lacunary sequences

In 1975 Walter Philipp proved the law of the iterated logarithm (LIL) for the discrepancy of lacunary sequences: for any sequence $(n_k)_{k \geq 1}$ satisfying the Hadamard gap condition $n_{k+1} / n_k \geq q > 1,~k \geq 1,$ we have $$ \frac{1}{4 \sqrt{2}} \leq \limsup_{N \to \infty} \frac{N D_N(\{ n_1 x \}, \dots, \{n_N x\})}{\sqrt{2 N \log \log N}} \leq C_q $$ for almost all $x$. In recent years there has been significant progress concerning the precise value of the limsup in this LIL for special sequences $(n_k)_{k \geq 1}$ having a ``simple'' number-theoretic structure. However, since the …