The Distribution of Totients

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Abstract

This paper is a comprehensive study of 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 ≥x, A(m), the number of solutions of Φ(x) = m (the “multiplicity” of m), and V k (x), the number of mx 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 then V k (x) ≫ V(x). Sierpiński conjectured that every multiplicity k ≥ 2 is possible, and we deduce this from the Prime k-tuples Conjecture. 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. The lower bound for a possible counterexample is extended to (math) and the bound lim inf x→∞ V 1(x)/V(x) ≤ 10-5.000.000.000 is shown. Determining the order of V(x) and V k (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 logx, where c ≈ 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.

Dedicated to the memory of Paul Erdős