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13
The distribution of totients
, 1998
"... This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(x), the numb ..."
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Cited by 15 (6 self)
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This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(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 of V (x). It is also shown that for each k ≥ 1, if there is a totient with multiplicity k, thenVk(x)≫V(x). We further show that every multiplicity k ≥ 2 is possible, settling an old conjecture of Sierpiński. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of V (x) andVk(x) also provides a description of the “normal ” multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a preimage of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x is c log log x, wherec≈2.186. Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler’s function.
The number of solutions of Φ(x) = m
"... 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, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an uncondit ..."
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Cited by 9 (2 self)
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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, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 4 (3 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(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 φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
Values of the Euler Function in Various Sequences
 MONATSH. MATH. 146, 1–19
, 2005
"... Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as wel ..."
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Cited by 4 (3 self)
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Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as well as those positive integers n such that the equation ’ðnÞ f ðmÞ holds with some integer m, where f is a fixed polynomial with integer coefficients and degree deg f> 1.
PRIME CHAINS AND PRATT TREES
, 2009
"... We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x ..."
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Cited by 1 (0 self)
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We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x.
A weighted Turán sieve method
, 2006
"... We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients. ..."
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We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients.
SETS OF MONOTONICITY FOR EULER’S TOTIENT FUNCTION
"... Abstract. We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author. 1. ..."
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Abstract. We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author. 1.
ON COMMON VALUES OF φ(n) AND σ(m), II
"... Abstract. For each positiveinteger valued arithmetic function f, let Vf ⊂ N denote the image of f, and put Vf(x): = Vf ∩ [1,x] and Vf(x): = #Vf(x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sumofdivisors functio ..."
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Abstract. For each positiveinteger valued arithmetic function f, let Vf ⊂ N denote the image of f, and put Vf(x): = Vf ∩ [1,x] and Vf(x): = #Vf(x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sumofdivisors function. Work of Ford shows that Vφ(x) ≍ Vσ(x) as x → ∞. Here we prove a result complementary to that of Ford et al., by showing that most φvalues are not σvalues, and vice versa. More precisely, we prove that as x → ∞, #{n � x: n ∈ Vφ ∩ Vσ} � Vφ(x) + Vσ(x)
ON COMMON VALUES OF φ(n) AND σ(m), I
"... Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1. ..."
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Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1.
A VARIANT OF THE BOMBIERIVINOGRADOV THEOREM WITH EXPLICIT CONSTANTS AND APPLICATIONS
"... ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of t ..."
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ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of the BombieriVinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes. For integers a and q ≥ 1, let 1.