Kevin Ford, Paul Pollack
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.
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We prove unconditionally that for each $\ell \geq 1$, the difference $ϕ(p-\ell) - ϕ(p+\ell)$ is positive for $50\%$ of odd primes $p$ and negative for $50\%$.
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 …
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…
Using a remainder theorem for valuations of a field, we give a new perspective on the norm function of a global field. We define the Euler totient function of a global field and recover the essential analytical properties of the classical arithmetical function, namely the product formula. In addition, we prove the holomorphicity of the associated zeta function. As an application, we recover the analog of the mean value theorem of Erdős, Dressler, and Bateman via the Weiner-Ikehara theorem.
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…
We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $ϕ(n)$ and the sum-of-proper-divisors function $s(n):=σ(n)-n$. We prove that the values $ϕ(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where th…
Distribution of values of general Euler totient function
本文研究一般欧拉函数 $Φ_k(n)$ 的分布，证明了任意 $k\ge 1$ 时的联合分布结果，并展示了其极值阶。
We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence of a positive density of odd numbers m, such that for any $s\ge0$ and any even number $a$, the residue class $a\pmod{2^sm}$ contains infinitely many totients.
We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $ϕ(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as $k\to\infty$. Here, $ϕ(n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $λ(n)$ and the least such $n$ satisfies $n\ll m^{13}$.
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…
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。
Rank Amplification for Shifted Equal Values of Euler's Totient Function
研究欧拉函数在移位整数上取等值的计数问题，通过结合光滑函数定理与标记供应系统等新方法，得到单位移位及均匀移位范围的渐近公式与上界估计。
The Forestry of Adversarial Totient Iterations
该研究给出嵌套欧拉函数表达式的闭式解，拓展研究完全平方、立方序列对应的嵌套欧拉函数取值并证明其有界性，还提出可借助树状结构求解闭式的Arboreal Algorithm。
Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $φ$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set $\mathcal{V}$ has a positive upper Banach density.
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 the mean values of products of Dirichlet $L$-functions at positive integers
本文研究Dirichlet $L$-函数在正整数处乘积的均值分布，给出了由Dirichlet特征加权、涉及Bernoulli函数和Jordan totient函数的显式公式，推广了多种已知结果。
We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.
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 …
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).
Prime-Detecting Identities from Dirichlet Inversion
本文通过Dirichlet逆与Jordan totient函数构造恒等式，完全刻画其零点集，证明$F_1(n)=0$当且仅当$n$为素数或$n=18$，且$k\geq2$时$F_k(n)=0$当且仅当$n$为素数，并给出伴随刻画。
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.
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.
Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function $ϕ$ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers $a$ and $b$, the least positive integer $c=c(a,b)$ such that the quotient $ϕ(c!)/ϕ(a!)ϕ(b!)$ is an integer. We derive results on the limit of the ratio $c(a,b)/(a+b)$ as $a$ and $b$ tend to infinity. Furthermore, we show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ with an exception of a set of density zero.
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.
We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $φ(n)^r$, $σ(n)^r$ ($r>-1$ real), where $φ$ is Euler's totient function and $σ$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, sta…
We prove that for a large class of functions $P$ and $Q$, there exists $d\in (0,1)$ such that the discrete bilinear Radon transform $$B^{\rm dis}_{P,Q}(f,g)(n)=\sum_{m\in\mathbb{Z}\setminus\{0\}} f(n-P(m))g(n-Q(m))\frac{1}{m}$$ is bounded from $l^2\times l^2$ into $l^{1+ε}$ for any $ε\in (d,1)$. In particular, the boundedness holds for any $ε\in (0,1)$ when $P$ (or $Q$) is the Euler totient function $ϕ(|m|)$ or the prime counting function $π(|m|)$.
The frequency of occurrence of "locally repeated" values of arithmetic functions is a common theme in analytic number theory, for instance in the Erdős-Mirsky problem on coincidences of the divisor function at consecutive integers, the analogous problem for the Euler totient function, and the quantitative conjectures of Erdős, Pomerance and Sarkőzy and of Graham, Holt and Pomerance on the frequency of occurrences. In this paper we introduce the corresponding problems in the setting of polynomials over a finite field, and completely solve them in the large finite field limit.
For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.
Mean values of multiplicative functions and applications to residue-class distribution
本文给出了乘性函数部分和的一致界，并据此对 Alladi-Erdős 函数和 Euler totient 函数在不同模数下的剩余类分布给出了近乎最优的估计与均匀性量化分析。
Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕdenote Euler's totient function, let σbe the sum-of-divisors function, and let λbe Carmichael's lambda-function. We show that if f is any function formed by composing ϕ, σ, or λ, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base g digits of successive f-values is g-normal. We also prove the same result if the inputs 1, 2, 3, \dots are replaced with the primes 2, 3, 5, \dots. The proof is an adaptation of …
The range and omitted values of a certain sequence involving the partition function
研究分拆函数p(n)与n之差构成数列的值域，给出组合解释，证明其稀疏性（计数函数阶为(log x)^2，自然密度为零），并扩展到Durfee正方形约束的分拆。
Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $ζ^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $ζ^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the o…
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