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.

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.

The book under review gives a comprehensive account of the Rosser-Iwaniec method, the most important development in the construction of number sieves since the advent in 1947 of Selberg’s λ-method. 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