this paper, we shall use the following notations: p i for the i-th prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) (this is the Liouville function) and (n) = ( 1) !(n) so that (n) is completely multiplicative and (n) is multiplicative, and let LN = f(1); (2); : : : ; (N)g GN = f(1); (2); : : : ; (N)g: For y 1 let y (n) and y (n) denote the multiplicative functions de ned by y (p ( 1) (= (p +1 for p > y y (p 1 (= (p +1 for p > y; respectively, and write LN (y) = f y (1); y (2); : : : ; y (N)g GN (y) = f y (1); y (2); : : : ; y (N)g: Research partially supported by Hungarian National Foundation for Scienti c Research, Grant No. T017433 MKM fund FKFP-0139/1997 and by French-Hungarian APAPE-OMFB exchange program F5 /97

This paper is an announcement of many new results concerning 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 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 pre-image 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.

Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].

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.

Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity

This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and

Abstract. Numbers of the form Fb,n = b2n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m −1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn’s quantitative form of “Hypothesis H” of Schinzel and Sierpiński. A list of the current largest known GFN primes is included. 1.

this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is well-known (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n

We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and k-tuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and k-tuples of primes separated by fixed gaps. For example, if d is even, we define P d (x) = |{0 < n < x : n, n + d are both prime}|. They conjecture both that lim x## P d (x) P 2 (x) = # odd p|d p - 1 p - 2 and that P 2 (x) is asymptotic to 2 # p>2 # 1 - 1 (p - 1) 2 # # x 2 dy (log y) 2 . We will refer to the first equation as the "relative conjecture" and the second as the "absolute conjecture." There has been much numerical verification of these conjectures, and many attempts at proofs. Balog [1] proves a result that implies that the conjectures are true "on average," where the average is taken over the possible shapes of the k-tuples. Golubev [6] compares these conjectures with provable analogous limit results for patterns of numbers prime to n. Turan [18] relates such theorems to zeroes of the #-function, using the large sieve rather than Hardy and Littlewood's circle method. There are also many generalizations to specific fields. Most of those generalizations use "Conjecture H" of Sierpinski and Schinzel [14,15]. For example, Sierpinski [17] shows that Conjecture H implies the existence of infinitely many prime Gaussian integers di#ering by 2. Bateman and Horn [2,3] quote a quantitative form of Conjecture H which allows them to estimate the density of rational twin primes. Shanks [16] numerically verifies that the density of prime pairs of the form a + i, a + 2 + i in the Gaussian integers matches that of the quantitative form of Conjecture H. Rieger ...

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 sum-of-divisors function. This proves a 50-year 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 Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.