Search

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

Conjecture H of Hardy and Littlewood says that every large integer is either a square or the sum of a prime and a square. Despite progress showing that this is true for almost all large integers, Conjecture H remains unproven. In this paper, Conjecture H is approximated using Piatetski–Shapiro sequences by showing that all large integers can be written in the form p + [ m c ] for any fixed 1 < c < 6 9 / 6 2. The existing results for the conjecture are also extended by showing that almost all large integers can be written in the form [ p c ] + m 2 for any fixed 1 < c < 4 1 9 / 3 5 2. [ABSTRACT FROM AUTHOR]

The Piatetski-Shapiro sequences are sequences of the form ( n c ) with c ≥ 1 and c ∉ ℕ. We investigate the distribution of k th powers in Piatetski-Shapiro sequences. Moreover, we study the solution of a more general equation n c = s m k with m , n ∈ ℕ , 1 ≤ n ≤ N for an integer s. We obtain bounds for different values of k and an average bound over k -free numbers s in an interval. In the end, we also study the solution of the equation n c = s p 2 for which p is a prime number. [ABSTRACT FROM AUTHOR]

Let f (t) be an arbitrary real-valued positive nondecreasing function, in this paper we prove that S f , k (x) − T f , k (x) = 𝒪 (f (x) x 1 3 1 / 4 1 6 (log x) 2 6 9 4 7 / 8 3 2 0) , S f , k (x) − T f , k (x) = S f , 1 (x) − T f , 1 (x) + 𝒪 (f (x) x 2 2 7 / 7 9 6 + 𝜀) , where S f , k (x) is the sum given in the title, T f , k (x) = ∫ 1 x f (t) { x t } k d t and k is a positive integer. This improves the result of Mercier and Nowak. [ABSTRACT FROM AUTHOR]

An integer n ≥ 1 is said to be practical if every natural number m ≤ n can be expressed as a sum of distinct positive divisors of n. The number of practical numbers up to x is asymptotic to c x / log x , where c is a constant. In this paper, we show that c = 1. 3 3 6 0 7 .... [ABSTRACT FROM AUTHOR]

Let [ 𝜃 ] denote the integral part of the real number 𝜃. In this paper, it is proved that for 2 < c < 4 0 8 1 9 7 , the Diophantine equation [ p 1 c ] + [ p 2 c ] + [ p 3 c ] + [ p 4 c ] + [ p 5 c ] = N is solvable in prime variables p 1 , ... , p 5 for sufficiently large integer N. [ABSTRACT FROM AUTHOR]

Over the ring of algebraic integers 𝒪 of a number field K , the quadratic congruence a 1 x 1 2 + ⋯ + a k x k 2 ≡ c modulo a nonzero ideal ℐ is considered. We prove explicit formulas for N k (a 1 , ... , a k , c , ℐ) and N k ∗ (a 1 , ... , a k , c , ℐ) , the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k and the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k with x 1 x 2 ⋯ x k coprime to ℐ , respectively. If ℐ is contained in a prime ideal 𝒫 containing the rational prime p , it is assumed that 𝒫 is unramified over p. Moreover, some interesting identities for exponential sums are proved. [ABSTRACT FROM AUTHOR]

Let denote the set of all nonnegative integers and be a subset of . The set is called an asymptotic basis of order if every sufficiently large integer can be written as the sum of two elements of . Otherwise, is called an asymptotic nonbasis of order . Let denote the number of representations of in the form , where and . An asymptotic nonbasis of order is called a maximal asymptotic nonbasis of order if is an asymptotic basis of order for every . In this paper, a maximal asymptotic nonbasis is constructed satisfying for all and as , where is an increasing sequence of . [ABSTRACT FROM AUTHOR]

In this paper, we study the hybrid problem of Hua's theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case , which deepen the classical result of Hua. [ABSTRACT FROM AUTHOR]

We obtain several estimates for double rational exponential sums modulo a prime with products where both and run through short intervals and is fixed integer. We also obtain some new estimates for the number of points on exponential modular curves and similar. [ABSTRACT FROM AUTHOR]

We use analytic methods to obtain an explicit formula for the fourth power mean where , is a Dirichlet character modulo and denotes the summation over all such that . This extends the result of Chen, Ai and Cai by overcoming the limitation . [ABSTRACT FROM AUTHOR]

For any positive integer , let be the ring of residue classes modulo and be the group of its units. Recently, for any , Yang and Tang obtained a formula for the number of solutions of the quadratic congruence with units, nonunits and mixed pairs, respectively. In this paper, for any , we give a formula for the number of representations of as the sum of two weighted squares of units modulo . We resolve a problem recently posed by Yang and Tang. [ABSTRACT FROM AUTHOR]

We consider the problem of finding small prime gaps in various sets . Following the work of Goldston-Pintz-Yıldırım, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting qn denote the nth prime in , we will establish that for any small constant ϵ > 0, the set {qn | qn+1 - qn ≤ ϵ n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao, we will also demonstrate that has bounded prime gaps. Specific examples, such as the case where is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets. [ABSTRACT FROM AUTHOR]

For each integer n ≥ 2, let be the index of composition of n, where γ(n) ≔ ∏p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑n≤xλ-k(n). Furthermore, we improve the error term under the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with $m=p_{1}^{r_{1}} \cdots p_{t}^{r_{t}}$ as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp(2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum \[ G_{{\it M}}(\chi)= \sum_{\alpha \in O{\!/\!}{\it M}^{*}} \chi(N\alpha) \zeta_{m}^{Tr \; \alpha} \] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums \[ R_{{\it M}}(\eta,z)= \sum_{\alpha\in O{\!/\!}{\it M}^{*} } \eta(N\alpha) \zeta_{m}^{Tr\,\alpha + z/N\alpha}, \quad (z \in {\bf Z}/m{\bf Z}^{*}) \] are well-defined for any numerical character η (mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2. [ABSTRACT FROM AUTHOR]