On sums of monotone functions over smooth numbers

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers ( A i ) i = 1 ∞ ⊆ d m n ( f ) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑ n ∈ A N P ( n ) ≤ N θ f ( n ) ∼ ρ ( 1 / θ ) ∑ n ∈ A N f ( n ) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

Quickselect and the Dickman Function

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.

Integers free of small prime factors in arithmetic progressions

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.