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For a function f : N → N , let N f + (x) = { n ⩽ x : n = k + f (k) for some k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ⩽ k ⩽ n : gcd ⁡ (k , n) = 1 } be Euler's totient function. We show that (1) x ≪ N ω + (x) , (2) x ≪ N τ + (x) ⩽ 0.94 x , (3) x ≪ N φ + (x) ⩽ 0.93 x. [ABSTRACT FROM AUTHOR]

Following V. I. Arnold, we define the stochasticity parameter |$S(U)$| of a subset |$U$| of |${\mathbb {Z}}/M{\mathbb {Z}}$| to be the sum of squares of the consecutive distances between elements of |$U$|⁠. In this paper, we study the stochasticity parameter of the set |$R_{M}$| of quadratic residues modulo |$M$|⁠. We present a method that allows to find the asymptotics of |$S(R_{M})$| for a set of |$M$| of positive density. In particular, we obtain the following two corollaries. Denote by |$s(k)=s(k,{\mathbb {Z}}/M{\mathbb {Z}})$| the average value of |$S(U)$| over all subsets |$U\subseteq {\mathbb {Z}}/M{\mathbb {Z}}$| of size |$k$|⁠ , which can be thought of as the stochasticity parameter of a random set of size |$k$|⁠. Let |${\mathfrak {S}}(R_{M})=S(R_{M})/s(|R_{M}|)$|⁠. We show that (a) |$\varliminf _{M\to \infty }{\mathfrak {S}}(R_{M})<1<\varlimsup _{M\to \infty }{\mathfrak {S}}(R_{M})$|⁠ ; (b) the set |$\{ M\in {\mathbb {N}}: {\mathfrak {S}}(R_{M})<1 \}$| has positive lower density. [ABSTRACT FROM AUTHOR]

For a function f : ℕ → ℕ , define N f × (x) = # { n ≤ x : n = k f (k)  for some  k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ≤ k ≤ n : (k , n) = 1 } be Euler's totient function. We prove that (1) N τ × (x) ≍ x (log x) 1 / 2 ; (2) N ω × (x) = (1 + o (1)) x log log x ; (3) N φ × (x) = (c 0 + o (1)) x 1 / 2 , where c 0 = 1. 3 6 5 .... [ABSTRACT FROM AUTHOR]

We prove that if ε(m) → 0 arbitrarily slowly, then for almost all m and any A ⊂ Zm such that A−A does not contain non-zero quadratic residues we have |A| ≤ m1/2−ε(m). [ABSTRACT FROM AUTHOR]