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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]

We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory. [ABSTRACT FROM AUTHOR]

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris 360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

In this paper, we study the Newton polygon of the L -function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the p -adic complex associated to the L -function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the average ∑ f ∈ κ ∗ (P) ω f − 1 ℒ f (α , β ; N) , where ℒ f (α , β ; N) = ∑ n ∼ N λ f (n) e (n 2 α + n β) for any α , β ∈ ℝ and N ≥ 2 , with f ∈ κ ∗ (P) running over primitive holomorphic cusp forms of weight κ and prime level P. As a result, we prove pointwise uniform bounds with respect to α , β for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum ℒ f (α , β ; N) , so long as the level of f varies. Concerning the applications of the main result, we consider the quadratic Waring problem associated to the coefficients of cusp forms exhibiting considerable cancellations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that S τ k (x) = ∑ n ≤ x τ k x n = ∑ n = 1 ∞ τ k (n) n (n + 1) x + O (x 9 / 1 9 + ) , where is an arbitrary small positive constant and τ k (n) is the number of representations of n as product of k natural numbers. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form for SL 2 (ℤ) with Laplace eigenvalue λ f (Δ) = 1 / 4 + μ 2 and let λ f (n) be its n th normalized Fourier coefficient. It is proved that, uniformly in α , β ∈ ℝ , ∑ n ≤ X λ f (n) e (α n 2 + β n) ≪ X 7 / 8 + λ f (Δ) 1 / 2 + , where the implied constant depends only on . We also consider the summation function of λ f (n) and under the Ramanujan conjecture we are able to prove ∑ n ≤ X λ f (n) ≪ X 1 / 3 + λ f (Δ) 4 / 9 + with the implied constant depending only on . [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]

We study sums of the shape ∑ n ≤ x f (⌊ x / n ⌋) where f is either the von Mangoldt function or the Dirichlet–Piltz divisor functions. We improve previous estimates when f = Λ and f = τ , and provide new results when f = τ r with r ≥ 3 , breaking the 1 2 -barrier in each case. The functions f = μ 2 , f = 2 ω and f = ω are also investigated. [ABSTRACT FROM AUTHOR]

Let f be any arithmetic function and define S f (x) : = ∑ n ≤ x f ([ x / n ]). If the function f is small, namely, f (n) ≪ n , then the error term in the asymptotic formula of S f (x) has the form O (x 1 / 2 + ). In this paper, we shall prove that if a small arithmetic function f has a binary additive property, then we can break the barrier 1 / 2 in the asymptotic formula of S f (x). [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]

Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on the earlier work by Madritsch and Tichy. In particular, let f = P + ϕ where P is a polynomial of degree k and ϕ is a linear combination of functions of shape x c , c ∉ ℕ , 1 < c < k. We prove that for any given irrational ξ we have min 2 ≤ p ≤ X p prime ∥ ξ ⌊ f (p) ⌋ ∥ ≪ f , X − ρ (k) + , for P belonging to a certain class of polynomials and with ρ (k) > 0 being an explicitly given rational function in k. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y (t) = ∑ j = 1 M (∑ m = 0 n j γ j , m t m) e 2 π λ j t , where the frequency parameters λ j ∈ ℂ are pairwise distinct. In order to reconstruct y(t) we employ a finite set of classical Fourier coefficients of y with regard to a finite interval (0 , P) ⊂ ℝ with P > 0. For our method, 2N + 2 Fourier coefficients c k (y) are sufficient to recover all parameters of y, where N : = ∑ j = 1 M (1 + n j) denotes the order of y(t). The recovery is based on the observation that for λ j ∉ i P ℤ the terms of y(t) possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput.40(3) (2018) A1494A1522]. If a sufficiently large set of L Fourier coefficients of y is available (i.e. L ≥ 2 N + 2), then our recovery method automatically detects the number M of terms of y, the multiplicities n j for j = 1 , ... , M , as well as all parameters λ j , j = 1 , ... , M , and γ j , m , j = 1 , ... , M , m = 0 , ... , n j , determining y(t). Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. [ABSTRACT FROM AUTHOR]

Let q [ t ] be the polynomial ring over the finite field q of q elements. For a natural number N , let N be the set of all polynomials in q [ t ] of degree less than N. Let h be a quadratic polynomial over q [ t ]. Suppose that h is intersective, that is, which satisfies (A − A) ∩ (h ( q [ t ]) ∖ { 0 }) ≠ ∅ for any A ⊆ q [ t ] with limsup N → ∞ | A ∩ N | / q N > 0 , where A − A denotes the difference set of A. Let B ⊆ N. Suppose that (B − B) ∩ h ( q [ t ]) ∖ { 0 } = ∅ and that the characteristic of q is not divisible by 2. It is proved that | B | ≤ C N − c log log log N q N for any 0 < c < 1 / log 3 , where C ≥ 1 is a constant depending only on q , h and c. [ABSTRACT FROM AUTHOR]

We investigate the order of exponential sums involving the coefficients of general L -functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for G L m (ℤ) , Sci. China Math.58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math.57 (2017) 185–204]. [ABSTRACT FROM AUTHOR]

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial t -designs have attracted lots of research interest for decades. The interplay between coding theory and t -designs started many years ago. It is generally known that t -designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a t -design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of 2 -designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of 2 -designs are calculated explicitly. [ABSTRACT FROM AUTHOR]

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms. [ABSTRACT FROM AUTHOR]

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory. [ABSTRACT FROM AUTHOR]

In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability. [ABSTRACT FROM AUTHOR]

The local sum conjecture is a variant of some of Igusa's questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko's treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective. [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]

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials. [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]

In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients. [ABSTRACT FROM AUTHOR]

We investigate the existence of large values of L -functions attached to Maass forms on the critical line with prescribed argument. The results obtained rely on the resonance method developed by Soundararajan and furthered by Hough. [ABSTRACT FROM AUTHOR]

Let μ { p n , d n } be a Cantor–Moran measure given by the infinite convolution of finite measures with equal probability μ { p n , d n } = δ p 1 − 1 { 0 , d 1 } ∗ δ (p 1 p 2) − 1 { 0 , d 2 } ∗ ⋯ , where { p n , d n } ⊂ ℤ and 0 < d n < p n for n ≥ 1. In this paper, we present a complete characterization for maximal orthogonal sets of exponentials of L 2 (μ { p n , d n }) in terms of maximal mappings. As its application, we give a sufficient condition for a maximal orthogonal set to be a basis. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime and let 𝔽 q be the finite field consisting of q elements. We establish new explicit estimates on Gauss sums of the form S n (a) = ∑ x ∈ 𝔽 q ψ a (x n) , where ψ a is a nontrivial additive character. In particular, we show that one has a nontrivial upper bound on | S n (a) | for certain values of n of order up to q 1 / 2 + 1 / 6 8 . Our results improve on the previous best-known bound due to Zhelezov. [ABSTRACT FROM AUTHOR]

We fix a monic polynomial f (x) ∈ 𝔽 q [ x ] over a finite field of characteristic p of degree relatively prime to p. Let a ↦ ω (a) be the Teichmüller lift of 𝔽 q , and let χ : ℤ p → ℂ p × be a finite character of ℤ p . The L -function associated to the polynomial f and the so-called twisted character ω u × χ is denoted by L f (ω u , χ , s) (see Definition 1.2). We prove that, when the conductor of the character is large enough, the p -adic Newton slopes of this L -function form arithmetic progressions. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large real number. In this paper, it is proved that, for 1 < c < 1 1 9 3 8 8 9 , the following Diophantine inequality | p 1 c + p 2 c + p 3 c + p 4 c − N | < log − 1 N is solvable in prime variables p 1 , p 2 , p 3 , p 4 , which improves the result of Mu [On a Diophantine inequality over primes, Adv. Math. (China)44(4) (2015) 621–637]. [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]

We estimate the exponential sum ∑ 1 ≤ n ≤ x e (α n + f 1 (n) + ⋯ + f ℓ (n)) , where α is a real number and f 1 , ... , f ℓ are digital functions; in the spirit of the works of Kim and Berend–Kolesnik. A similar estimate along short intervals is also provided. [ABSTRACT FROM AUTHOR]

Hubert, Mauduit and Sárközy introduced pseudorandom measures for finite pseudorandom binary lattices. Gyarmati, Mauduit, Sárközy and Stewart presented some natural and flexible constructions, which are the two-dimensional extensions and modifications of a few one-dimensional constructions. The upper estimates for the pseudorandom measures of their binary lattices are based on the principle that character sums or exponential sums in two variables can be estimated by fixing one of the variables. In this paper, we constructed two large families of n dimensional pseudorandom binary lattices by using the multiplicative inverse modulo p , and study the properties: pseudorandom measure, collision and avalanche effect. [ABSTRACT FROM AUTHOR]

We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions. [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]

In this paper, we consider the mean-value estimate for nonlinear Weyl sums over primes in short intervals and establish the related Bombieri-type theorems. These results have applications in additive problems with prime variables in short intervals. [ABSTRACT FROM AUTHOR]

Let 1 < c < 4 3 3 6 . In this paper, it is proved that for every sufficiently large real number N , the Diophantine inequality | p 1 c + p 2 c + p 3 c − N | < N − 9 1 0 c (4 3 3 6 − c) is solvable in primes p 1 , p 2 , p 3 . This result constitutes an improvement upon that of Baker and Weingartner. [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]

For and , , let denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs , , with tending to infinity as grows, such that for all . [ABSTRACT FROM AUTHOR]

In this paper, we prove a multiple Abel summation formula and give some applications of this formula. We use these evaluations in tandem with some recent results to obtain various asymptotic evaluations for multiple sums. [ABSTRACT FROM AUTHOR]

In 1862, Wolstenholme proved that the numerator of the th harmonic number is divisible by for any prime . A variation of this theorem was shown by Alkan and Leudesdorf. Motivated by these results, we prove a congruence modulo some odd primes for some generalized harmonic type sums. [ABSTRACT FROM AUTHOR]

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on . We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve. [ABSTRACT FROM AUTHOR]

Let be a Dirichlet character (mod ) with conductor . In a quite recent paper Zhao and Cao deduced the identity , which reduces to Menon's identity if is the principal character (mod ). We generalize the above identity by considering even functions (mod ), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums. [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]