# The Distribution of Totients

@article{Ford1998TheDO,
title={The Distribution of Totients},
author={Kevin Ford},
journal={The Ramanujan Journal},
year={1998},
volume={2},
pages={67-151}
}
• K. Ford
• Published 1 March 1998
• Mathematics
• The Ramanujan Journal
AbstractThis paper is a comprehensive study of the set of totients, i.e., the set of values taken by Euler's φ-function. The main functions studied are V(x), the number of totients ≤x, A(m), the number of solutions of φ(x)=m (the “multiplicity” of m), and V_k(x), the number of m ≤ x with A(m)=k. The first of the main results of the paper is a determination of the true order ofV(x) . It is also shown that for each k ≥ 1, if there is a totient with multiplicity k then V_k(x) ≫ V(x). Sierpiński…

## Tables from this paper

The distribution of totients
This paper is a comprehensive study of the set of totients, i.e., the set of values taken by Euler’s Φ-function. The main functions studied are V(x), the number of totients ≥x, A(m), the number of
On the distribution of totients
An integer n is called a totient if there is some integer x such that <p(x) = n, where <p is Euler's function. If this equation is not solvable, n is called a nontotient. In 1956, Schinzel [4] proved
An arithmetic function arising from Carmichael’s conjecture
• Mathematics
• 2011
Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n, the equation φ(n) = φ(m) has a solution m 6= n. This suggests defining F (n) as the number of
The number of solutions of q (x) = m
An old conjecture of Sierpiniski asserts that for every integer k ) 2, there is a number m for which the equation ((x) = m has exactly k solutions. Here q is Euler's totient function. In 1961,
Common values of the arithmetic functions ϕ and σ
• Mathematics
• 2010
We show that the equation ϕ(a) = σ(b) has infinitely many solutions, where ϕ is Euler's totient function and σ is the sum‐of‐divisors function. This proves a fifty‐year‐old conjecture of Erdős.
The Number of Solutions of φ (x) = m
An old conjecture of Sierpinski asserts that for every integer k 2, there is a number m for which the equation (x )= m has exactly k solutions. Here  is Euler's totient function. In 1961, Schinzel
On the distribution of totients 2 mod. 4
• Mathematics
• 2018
In this paper we study the distribution of totients $2$ mod. $4$. We prove that the asymptotic magnitude of such totients with multiplicity two is half of that of prime numbers. As a corollary we
The number of solutions of φ ( x )
• Mathematics
• 2000
An old conjecture of Sierpiński asserts that for every integer k > 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961,
POPULAR SUBSETS FOR EULER’S φ-FUNCTION
Let φ(n) = #(Z/nZ)× (Euler’s totient function). Let > 0, and let α ∈ (0, 1). We prove that for all x > x0( , α) and every subset S of [1, x] with #S ≤ x1−α, the number of n ≤ x with φ(n) ∈ S is at

## References

SHOWING 1-10 OF 48 REFERENCES
On a conjecture of Carmichael
V. L. KLEE, JR. 1 Carmichael [ l ] 2 conjectured that for no integer n can the equation (x)=n ( being Euler's totient) have exactly one solution. To support the conjecture, he showed that each n for
The Number of Solutions of φ (x) = m
An old conjecture of Sierpinski asserts that for every integer k 2, there is a number m for which the equation (x )= m has exactly k solutions. Here  is Euler's totient function. In 1961, Schinzel
Some remarks on Euler's $\phi$ function and some related problems
The function <i>(n) is defined to be the number of integers relatively prime to n, and <t>(n)~n'JjLp\n(l—p~~)In a previous paper I proved the following results : (1) The number of integers w i « for
On two conjectures of Sierpiski concerning the arithmetic functions $and$
• Mathematics
• 1999
Let σ(n) denote the sum of the positive divisors of n. In this note it is shown that for any positive integer k, there is a number m for which the equation σ(x) = m has exactly k solutions, settling
A heuristic asymptotic formula concerning the distribution of prime numbers
• Mathematics
• 1962
Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these
Carmichael's conjecture on the Euler function is valid below 10 10,000,000
• Mathematics
• 1994
CarmichaeFs conjecture states that if ix) = n , then (y) = n for some y ^ x (^ is Euler's totient function). We show that the conjecture is valid for all x under io10'900'000 . The main new idea is
On Euler's $\phi$-function
For m > 1, ϕ(m) = |Z * m | = number of numbers less than m and relatively prime to m For all p, if p is prime, then ϕ(p) = p − 1.
On the values of Euler's φ-function
• Mathematics
• 1973
Introduction. Let-if denote the set of distinct values of filer's T-function, that is,)AE lh1 if and only if
Integers without large prime factors
• Mathematics
• 1993
© Université Bordeaux 1, 1993, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions