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31
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.
NonAbelian Generalizations of the ErdősKac Theorem
, 2001
"... Abstract. Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance: The number of n ≤ x coprime to a sati ..."
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Cited by 7 (5 self)
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Abstract. Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance: The number of n ≤ x coprime to a satisfying
The prime factors of Wendt's Binomial Circulant Determinant
 Math. Comp
, 1991
"... : Wm , Wendt's binomial circulant determinant, is the determinant of an m by m circulant matrix of integers, with (i; j)th entry i m ji\Gammajj j whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m 200 by implementing a new method for computati ..."
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Cited by 6 (0 self)
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: Wm , Wendt's binomial circulant determinant, is the determinant of an m by m circulant matrix of integers, with (i; j)th entry i m ji\Gammajj j whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = mp + 1 are odd primes, 3 does not divide m and m 200, then the first case of Fermat's Last Theorem is true for exponent p. 1. Introduction. For a given positive even integer m, define Wm to be the determinant of the m by m circulant matrix with top row (a 0 ; a 1 ; : : : ; am\Gamma1 ) where gm (X) := m\Gamma1 X i=0 a i X i := 8 ! : (X + 1) m \Gamma X m if 6 does not divide m; (X+1) m \GammaX m (X 2 +X+1) if 6 divides m. When 6 does not divide m, the (i; j)th entry is i m ji\Gammajj j and this matrix is given the name in the title. There are a variety of app...
Sieving and the ErdősKac Theorem
, 2006
"... We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. ..."
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Cited by 5 (0 self)
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We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.
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.
On multiplicatively perfect numbers
 J. Inequal. Pure Appl. Math
"... ABSTRACT. We study multiplicatively perfect, superperfect and analogous numbers. Connection to various arithmetic functions is pointed out. New concepts, inequalities and asymptotic evaluations are introduced. Key words and phrases: Perfect numbers, arithmetic functions, inequalities in Number theor ..."
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Cited by 3 (2 self)
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ABSTRACT. We study multiplicatively perfect, superperfect and analogous numbers. Connection to various arithmetic functions is pointed out. New concepts, inequalities and asymptotic evaluations are introduced. Key words and phrases: Perfect numbers, arithmetic functions, inequalities in Number theory. 2000 Mathematics Subject Classification. 11A25,11N56, 26D15. 1.
Sieve methods in combinatorics
, 2005
"... We develop the Turán sieve and a ‘simple sieve ’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, vertexcolourings of graphs, Latin squares, connected graphs, and generators ..."
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Cited by 3 (1 self)
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We develop the Turán sieve and a ‘simple sieve ’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, vertexcolourings of graphs, Latin squares, connected graphs, and generators of groups. In addition, we give a spectral interpretation of the Turán sieve.
Limitations to the Equidistribution of Primes III
 Comp. Math
, 1992
"... : In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept f ..."
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: In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept fixed. However, by a new construction, we show herein that this fails in the same ranges, for a fixed and, indeed, for almost all a satisfying 0 ! jaj ! x= log N x. 1. Introduction. For any positive integer q and integer a coprime to q, we have the asymptotic formula (1:1) ß(x; q; a) ¸ ß(x) OE(q) as x ! 1, for the number ß(x; q; a) of primes p x with p j a (mod q), where ß(x) is the number of primes x, and OE is Euler's function. In fact (1.1) is known to hold uniformly for (1:2) q ! log N x and all (a; q) = 1, for every fixed N ? 0 (the SiegelWalfisz Theorem), for almost all q ! x 1=2 = log 2+" x and all (a; q) = 1 (the BombieriVinogradov Theorem) and for almost all q !...
The Arithmetic
"... We show that if we pick log d N integers in the interval [1 N ] then the probability that there is a subset of them whose product yields a perfect square is exp  c log N log log N for any c < 1 2d . The methods used to prove these results are elementary. 1. ..."
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We show that if we pick log d N integers in the interval [1 N ] then the probability that there is a subset of them whose product yields a perfect square is exp  c log N log log N for any c < 1 2d . The methods used to prove these results are elementary. 1.