We obtain an asymptotic formula for the number of square-free values among p À 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of square-free values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ¼ ðk þ oð1ÞÞ x ln 1Àa x ; where a ¼ 0:37395y is the Artin constant, and k ¼ 0:80328y is another absolute constant.

Abstract. Let a be a natural number greater than 1. Let f a (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 satisfyinga , as x tends to infinity.

In a recent paper [G. Yu, An upper bound for B 2 [g] sets, J. Number Theory 122 (1) (2007) 211-220] Gang Yu stated the following conjecture: Let {p i } ∞ i=1 be an arbitrary sequence of polynomials with increasing degrees and all coefficients in {0, 1}. If we denote by (#p i ) the number of non-zero coefficients of p i , and let M(p 2 i ) be the maximal coefficient of p 2 i , then Q := lim inf i→∞ deg(p i )M(p 2 i ) (#p i ) 2 1, ( * ) as long as (#p i ) = o(deg p i ), as i → ∞. We give an explicit example that shows why this last condition is necessary, and we investigate some open questions it suggests.

As usual let s = σ + it. For any fixed value t = t 0 with |t 0 | ≥ 8, and for σ ≤ 0, we show that |ζ(s)| is strictly monotone decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality relating the monotonicity of all three functions is proved: 1 2000 Mathematics Subject Classification : Primary 11M06, Secondary 11M26

Let wðnÞ denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n # x for which wðnÞ is the sum of two square numbers. We also obtain similar results for the Dedekind function cðnÞ and the sum of divisors function sðnÞ: