Search

We obtain some improved results for the exponential sum $\sum _{x with $\theta \in (0,5/12),$ where $\Lambda (n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Murty and Srinivas ['On the uniform distribution of certain sequences', Ramanujan J. 7 (2003), 185–192]. [ABSTRACT FROM AUTHOR]

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for f (x) = x k + x c where k is a positive integer, c > 1 is a non-integer and ξ a real number, they obtained min 2 ≤ p ≤ X ‖ ξ ⌊ f (p) ⌋ ‖ ≪ k , c , ϵ X - ρ 2 (c , k) + ϵ for ρ 2 (c , k) > 0 explicitly given. In the present note, we improve upon their results of the case c > k and c > 4 . [ABSTRACT FROM AUTHOR]

In this note, we obtain asymptotic results on integer parts of ap that are free of kth powers of primes, where p is a prime number and a is a positive real number. [ABSTRACT FROM AUTHOR]

In this paper, we treat the estimate on exponential sums over cubes of primes in short intervals, and improve a previous bound of Kumchev (in: Number Theory: Arithmetic in Shangri-La (Proc. China-Japan Seminar Number Theory), pp. 116–131, World Scientific, Singapore, 2013). Moreover, we present some applications to the cubic Waring–Goldbach problem. [ABSTRACT FROM AUTHOR]

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map x ↦ (a x + b) / (c x + d) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences. [ABSTRACT FROM AUTHOR]

In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3$\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi−(N/j)1/3|≤N1/3−δ+ε(1≤i≤j),$\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some 0

Let c > 1 and 0 < γ < 1 be real, with c ∉ N . We study the solubility of the Diophantine inequality p 1 c + p 2 c + ⋯ + p s c - N < ε in Piatetski-Shapiro primes p 1 , p 2 , ... , p s of index γ —that is, primes of the form p = ⌊ m 1 / γ ⌋ for some m ∈ N . [ABSTRACT FROM AUTHOR]

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions. [ABSTRACT FROM AUTHOR]

Let K be a positive integer, { A i , 1 ≤ i ≤ K } be any partition of the sequence of squares of primes and s ( K ) be the smallest positive integer such that every sufficiently large integer can be written as the sum of no more than s ( K ) elements, which belong to one of the sets A i . In this paper, we prove that s ( K ) ≪ ϵ K 2 + ϵ for sufficiently small positive number ϵ and all K ≥ 1 . [ABSTRACT FROM AUTHOR]

Let α, α, α, β, β, β be real numbers with α, α, α >1. Suppose that each individual α is of a finite type and that at least one pair $\alpha_{i}^{-1}$, $\alpha_{j}^{-1}$ is also of a finite type. In this paper we prove that every large odd integer n can be represented as with p= n/3+ O( n(log n)) and $p_{i}\in\mathcal{B}_{i}$, where c>0 is an absolute constant and $\mathcal{B}_{i}$ denotes the so-called Beatty sequence, i.e. [ABSTRACT FROM AUTHOR]

We continue the study of sums of the form begun by Indlekofer and Kátai. Here | Y n|,| X p| ≦ 1 and α is irrational. We prove one conjecture of Kátai, disprove another by both authors, and give what may be a close to best possible result valid for all irrational α. [ABSTRACT FROM AUTHOR]

In this paper, we prove that each sufficiently large integer N ≢ 1 (mod 3) can be written as , with where U = N 9/20 + ɛ and p, p j are primes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are able to sharpen Hua''s classical result by showing that each sufficiently large integer can be written as where and p, are primes. This result improves a previous result with replaced by . [Copyright &y& Elsevier]

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua’s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as , where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following estimate on exponential sums over primes: Let k ⩾ 1, β k = 1/2 + log k/log 2, x ⩾ 2 and α = a/ q + λ subject to ( a,q) = 1, 1 ⩽ a ⩽ q, and λ ∈ ℝ. Then As an application, we prove that with at most O( N 7/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]