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In this paper, we derived that a sufficiently large integer N can always be represented as the sum of three generalized D. H. Lehmer numbers. As a consequence, we deduced Lu and Yi's original result (Monatsh. Math., 159 (2010), 45-58). [ABSTRACT FROM AUTHOR]

In this part I discuss the role of computers in the current research on additive number theory, in particular, in the solution of the classical Waring problem. In its original 18th century form, this problem consisted in finding for each natural k the smallest such that all natural numbers can be written as sums of s nonnegative kth powers: . In the 19th century, the problem was modified as the quest of finding such a minimal that almost all can be expressed in this form. In the 20th century, this problem was further specified as finding such and the precise list of exceptions. The 20th century problem is still unsolved even for cubes. However, even the solution of the original Waring problem was [almost] finalized only in 1984, with heavy use of computers. In the present paper, we document the history of this classical problem and its solution, discuss the possibilities of using this and surrounding material in education, and cover some further related aspects. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of 2. This improves the previous result. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Denote by D1/k the least integer d such that for every set A of nonnegative integers with the lower density l/k, the set (k + l)A contains an infinite arithmetic progression with difference at most d, where (k +l)A is the set of all sums of k + l elements (not necessarily distinct) of A. Chen and Li (2019) conjectured that D1/k = k2 + o(k2). The purpose of this paper is to confirm the above conjecture. We also prove that D1/k is a prime for all sufficiently large integers k. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that every sufficiently large even number can be expressed as a pair of equations, each containing two prime squares, four prime cubes and 150 powers of 2. [ABSTRACT FROM AUTHOR]

Let dk(n) denote the k-th divisor function. In this paper, we give the asymptotic formula of the sum [ABSTRACT FROM AUTHOR]

Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider exceptional sets in the Waring–Goldbach problem for fifth powers. We obtain new estimates of E s (N) (12 ≤ s ≤ 20) , which denote the number of integers n ≤ N such that n ≡ s (mod 2) and n cannot be represented as the sum of s fifth powers of primes. For example, we prove that E 20 (N) ≪ N 1 - 1 4 - 27 1600 + ϵ for any ϵ > 0 . This improves upon the result of Feng and Liu (Front Math China 16:49–58, 2021). [ABSTRACT FROM AUTHOR]

In this paper, we will investigate three Goldbach–Linnik type problems on cubes of primes. For example, it was proved that, for k = 1364 , every pair of sufficiently large even integers can be represented by a pair of eight prime cubes and k powers of 2. In this paper, we give the detailed proof for the first time and sharpen the value of k to 658. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of the coefficients appearing in the q-series expansion of ∏ n ≥ 1 [ (1 - q n) / (1 - q pn) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p = 3 case. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every sufficiently large even integer can be represented as a sum of two squares of primes, four cubes of primes, and 21 powers of 2. [ABSTRACT FROM AUTHOR]

In this paper, we obtain when k = 26 , every sufficiently large even integer can be represented as a sum of two prime squares, four prime cubes and k powers of 2. What is more, when k = 58 , every sufficiently large odd integer can be represented as a sum of one prime, four prime cubes and k powers of 2. These improve previous results. [ABSTRACT FROM AUTHOR]

Let d k (n) denote the kth divisor function. In this paper, we proved that the sum ∑ 1 ≤ p 1 , p 2 , p 3 ≤ X d k (p 1 2 + p 2 2 + p 3 2) has an asymptotic formula, where p 1 , p 2 , p 3 are primes and k ≥ 2 is an integer. [ABSTRACT FROM AUTHOR]

In this paper, we will first establish a power saving result for the shifted convolution sums of k-th powers and the normalized Fourier coefficients of SL 2 (Z) cusp forms. Later we will generalize the result to higher rank cases. [ABSTRACT FROM AUTHOR]

Let d3(n) denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum ... with n1; n2; n3 2 Z+ and k > 3 be an integer. Previously only the case of k = 2 is studied. [ABSTRACT FROM AUTHOR]

It was proved that, for k = 3 4 1 , every sufficiently large even integer can be represented in the form of eight cubes of primes and k powers of 2. In this paper, we sharpen the value of k to 204. [ABSTRACT FROM AUTHOR]

In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$. In particular, we prove that $k=335$ is admissible, which improves the previous result. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O (N 4 / 27 + ε) exceptions, all even positive integers up to N can be represented in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 6 + p 6 6 , where p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are prime numbers. This gives a large improvement of a recent result O (N 127 / 288 + ε) due to Liu (Proc. Indian Acad. Sci. (Math. Sci.) 130(1) (2020) Article ID. 8). [ABSTRACT FROM AUTHOR]

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx. 22 (2010) 403–419]. [ABSTRACT FROM AUTHOR]

Let (m 1 , ... , m J) and (r 1 , ... , r J) be two sequences of J positive integers satisfying 1 ≤ r j < m j for all j = 1 , ... , J . Let (δ 1 , ... , δ J) be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product ∏ j = 1 J (∏ k ≥ 1 (1 - q r j + m j (k - 1) ) (1 - q - r j + m j k ) ) δ j. Our work generalizes many known results, including an asymptotic formula due to Lehner for the partition function arising from the first Rogers–Ramanujan identity. The main technique used here is based on Rademacher's circle method. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N119/270+s) exceptions, all even positive integers up to N can be represented in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 6 + p 6 6 , where p1, p2, p3, p4, p5, p6 are prime numbers. [ABSTRACT FROM AUTHOR]

In this paper, we obtain when k = 98 , every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of four prime squares and k powers of 2. This improves the previous result k = 142 of Hu and Liu. [ABSTRACT FROM AUTHOR]

In this paper, we investigate exceptional sets in the biquadratic Waring-Goldbach problem. For example, we prove that all but O (N 60 / 61) integers subject to the necessary local conditions can be represented as the sum of seven biquadrates of primes. Furthermore, the method in this paper can be easily used for exceptional sets for sums of fifth and higher of powers of primes. For a video summary of this paper, please visit https://youtu.be/PGa8P-Px8bk. [ABSTRACT FROM AUTHOR]

In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (Acta Arith. 85:99–118, 1998). [ABSTRACT FROM AUTHOR]

The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method" to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined for α ∈ Q by ∑ n = 0 ∞ p α (n) x n : = ∏ k = 1 ∞ (1 - x k) - α . In this paper we use the Rademacher circle method to find an exact formula for p α (n) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that p α (n) satisfies. [ABSTRACT FROM AUTHOR]

Let k ≥ 3 be an integer, s = 2 k + 1 , (a i , a j) = 1 , 1 ≤ i < j ≤ s , where a 1 , ... , a s are nonzero integers, and let n be an integer. Suppose that a 1 , ... , a s satisfy some necessary congruent conditions. In this paper, it is proved that (i) if a j are not all of the same sign, then the equation a 1 p 1 k + ⋯ + a s p s k = n has prime solutions satisfying p j ≪ | n | 1 / k + max { | a j | } C (k) + 𝜀 , (ii) if all a j are positive and n ≫ max { | a j | } k C (k) + 1 + 𝜀 , then a 1 p 1 k + ⋯ + a s p s k = n is solvable in primes p j , where C (k) = 1 + 6 ⋅ 2 k + 2 0 3 ⋅ 2 2 k − 1 + 3 ⋅ 2 k − 1 − 2 0 . Our result uses C (3) = 3 9 2 2 ≈ 1. 7 7 2 7 ⋯ as an improvement of the recent result C (3) = 2 due to Zhao (2016), and largely improves the results C (k) = 3 ⋅ 2 k − 1 for k ≥ 4 proved by Yang and Hu (2016). [ABSTRACT FROM AUTHOR]

Let d(n) denote the Dirichlet divisor function. Define S k (x , y) : = ∑ x - y < m 3 k ⩽ x + y x - y < m i 2 ⩽ x + y i = 1 , 2 d (m 1 2 + m 2 2 + m 3 k) , 3 ⩽ k ∈ N. Let y = x 1 - δ k + 4 ε with δ 3 = 2 15 , δ k = 1 k (2 k - 2 + 1) for 4 ⩽ k ⩽ 7 , and δ k = 1 k (k 2 - k + 1) for k ⩾ 8 . In this paper, we establish an asymptotic formula of S k (x , y) and prove that S k (x , y) = K 1 L 1 (x , y) + 2 (γ K 1 - K 2) L 2 (x , y) + O (y 3 x - 2 + 1 k - ε ) , where K j (j = 1 , 2) are two constants and L j (x , y) (j = 1 , 2) satisfying L 1 (x , y) ≍ y 3 x - 2 + 1 / k log x , L 2 (x , y) ≍ y 3 x - 2 + 1 / k . [ABSTRACT FROM AUTHOR]

In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for n large enough, the two quantities are different and which of the two is bigger depends on the parity of n. This solves a recent conjecture formulated by Bringmann and Mahlburg (2012). [ABSTRACT FROM AUTHOR]

In this paper, we prove that, under some conditions, the values taken by certain real linear combinations of one prime, two squares of primes and s powers of two can arbitrarily close to any real number if s is sufficiently large, which improves the previous result. This argument can also lead to an improvement of the case of real linear combinations of four squares of primes and a bounded number of powers of two. [ABSTRACT FROM AUTHOR]

Let f be a normalized primitive holomorphic cusp form of even integral weight k for the full modular group SL(2, ℤ), and denote its nth Fourier coefficient by λf (n). We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017). [ABSTRACT FROM AUTHOR]

In this paper, we obtain an asymptotic formula of the sum ... with .... Moreover, if ..., we obtain a more refined asymptotic formula of this sum. [ABSTRACT FROM AUTHOR]

In this paper we obtain asymptotic formulas for positive crank and rank moments for overpartitions. Moreover, we show that crank and rank moments are asymptotically equal while the difference is asymptotically positive. This indicates that there exist analogous higher ospt-functions for overpartitions, which we define. [ABSTRACT FROM AUTHOR]

In this short paper, we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two squares of primes and a k-th power of prime for any integer k≥3. Our results improve the recent results due to Brüdern (in: Sander, Steuding, Steuding (eds) From arithmetic to zeta-functions, Springer, Cham 2016). The similar method can be also applied to some related questions in this direction, and this can improve the previous results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the application of the circle method to the problem of an asymptotics of the weighted number of integer points on multidimensional hyperboloids of a special form. We prove the convergence and positivity of the singular series and obtain an asymptotic formula for the singular integral of this problem. Earlier, only estimates for the singular integral were known. [ABSTRACT FROM AUTHOR]

We prove new lp(Zd) bounds for discrete spherical averages in dimensions d ≥ 5. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if Aλf is the spherical average of f over the discrete sphere of radius λ, we have... for any lacunary sets of integers {λ² k}. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]

In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average. [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]