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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.
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.
On the genera of X0(N
 J. Number Theory
"... Abstract. Let g0(N) be the genus of the modular curve X0(N). We record several properties of the sequence {g0(N)}. Even though the average size of g0(N) is (1.25/π 2)N, a random positive integer has probability zero of being a value of g0(N). Also, if N is a random positive integer then g0(N) is odd ..."
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Cited by 3 (1 self)
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Abstract. Let g0(N) be the genus of the modular curve X0(N). We record several properties of the sequence {g0(N)}. Even though the average size of g0(N) is (1.25/π 2)N, a random positive integer has probability zero of being a value of g0(N). Also, if N is a random positive integer then g0(N) is odd with probability one. 1.
Compositions with the Euler and Carmichael Functions
"... Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1 ..."
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Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 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.