We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence αn+ β , n= 1,2, . . . , where α,β ∈R, and α > 0 is irrational. For example, we show that ∑ n N ω ( αn+ β )∼N log logN and ∑ n N (−1)Ω( αn+β ) = o(N), where Ω(k) and ω(k) denote the number of prime divisors of an integer k = 0 counted with and without multiplicities, respectively. © 2006 Elsevi...

‎let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎in this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎this nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{ext}_{mathcal{a}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{z}/p,mathbb{z}/p)$‎ ‎in ...

Fix α ∈ ( 0 , 1 <mml:mrow class="MJX-TeXAtom-ORD"...

1. First we introduce some notations. elements of the finite set S is denoted by ISI. We write ex = exp x. We denote the i th prime number by p i ,. v(n) denotes the number of distinct prime factors of n. mk is the smallest integer m for which v(1k» > k. P(n) is the greatest, p(n) the smallest prime factor of n. nk is the smallest integer n for which P(n+i) > k for all 1 < i <_ k. Mk > c I k 2 ...

Let R be a ring with Jacobson ideal J and center C. McCoy and Montgomery introduced the concept of a p-ring (p prime) as a ring R of characteristic p such that xp = x for all x in R. Thus, Boolean rings are simply 2-rings (p = 2). It readily follows that a p-ring (p prime) is simply a ring R of prime characteristic p such that R ⊆ N + Ep, where N = {0} and Ep = {x ∈ R : xp = x}. With this as mo...

Primality test, prime In this paper we prove the following simplification of Miller's primality criterion [2]. Theorem 1. Assume that for every integer d that is 1 mod 4 and either prime or the product of two primes, the L-function I= & (k/d) l kmS satisfies the generalized Riemann hypothesis, where {kid) denotes the Jacobi symbol, defined below. Let n be an odd integer, n > I , and write n-I-=...

Abstract Let $(Z_n)_{n\geq 0}$ be a critical branching process in random environment defined by Markov chain $(X_n)_{n\geq with values finite state space $\mathbb{X}$ . $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ the walk associated to , where $f_i$ is offspring generating function when $i \in \mathbb{X}$ Conditioned on event $\{ Z_n&gt;0\}$ we show nondegeneracy of limit law normalized numbe...