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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.
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.
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... The book under review gives a comprehensive account of the RosserIwaniec method, the most important development in the construction of number sieves since the advent in 1947 of Selberg’s λmethod. Sieve literature has grown prodigiously since the publication of [HR], and Dr. Greaves has had to make ..."
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The book under review gives a comprehensive account of the RosserIwaniec method, the most important development in the construction of number sieves since the advent in 1947 of Selberg’s λmethod. Sieve literature has grown prodigiously since the publication of [HR], and Dr. Greaves has had to make some difficult decisions on what to include and what to omit. On the whole his choices have been wise; students and experts alike will have much to learn from his careful presentation. If the book does not always make for easy reading, that is due largely to the nature of the subject: not only does sieve architecture rest on complicated combinatorial foundations, but these culminate nowadays in none too easy linear differential delay boundary value problems and also link up with results and techniques from modern analytic number theory. We begin with a brief description of sieve methods by setting the stage: Let P be a finite set of primes—usually this is an infinite, increasing sequence of primes truncated at some number z>2—and refer to P as a ‘sieve’. Let P denote the product of all the primes in P. In Selberg’s terminology, P is said to ‘sift out ’ an integer n if n is divisible by some prime p in P. Then, writing (n, P)forthehighest common factor of n and P, P sifts out n if and only if (n, P)> 1; by the same token, the indicator function of all integers n that are not sifted out by P is ∑ 1 when (n, P)=1
Does Ten Have a Friend?
, 806
"... Any positive integer n other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization ..."
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Any positive integer n other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of n. 1 The Abundancy Index For a positive integer n, the sum of the positive divisors of n is denoted σ(n); the ratio σ(n) n is known as the abundancy ratio or abundancy index of n, denoted I(n). A perfect number is a positive integer n satisfying I(n) = 2. Considering the milleniaold interest in perfect numbers and the (at least) centuriesold interest in the “abundancy ” of positive integers, it is somewhat surprising that study of the abundancy index seems to have flourished only relatively recently; see [2], [5], and [6], and the references there to earlier work. Interesting questions have been asked and answered: for instance, it is now known ([5] and [6]) that both the range of the function I and the complement of that range in the rational numbers are dense in the interval (1, ∞). Questions about another kind of density remain. Let, for x> 1, J(x) = I−1 ((x, ∞)) = {n  I(n)> x}; does the limit J(x) ∩ {1,...,N} f(x) = lim N→ ∞ N exist? If so, what can be said about the behavior of the nonincreasing function f? Is it continuous? Strictly decreasing? The open question about the abundancy index to be addressed here, stated in the title and explained in the next section, is not so exotic – in fact, it has a (1) ∗Keywords: abundancy ratio, abundancy index, sum of divisors, perfect numbers, friendly