# 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} }

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…

## 75 Citations

The distribution of totients

- Mathematics
- 1998

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

- Mathematics
- 2002

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

- Mathematics
- 1999

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

- Mathematics
- 1999

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

- Mathematics
- 2018

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…

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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…

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