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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 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]

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, 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 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]

In this paper, we show that every sufficiently large even integer can be represented as eight prime cubes and k 1 powers of 2, and every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k 2 powers of 2, where k 1 = 28 and k 2 = 98. These results improve previous results k 1 = 30 and k 2 = 335 , respectively. [ABSTRACT FROM AUTHOR]

Let d k (n) denote the k -th divisor function. In this paper, we study the asymptotic formula of the sum ∑ 1 ≤ n 1 , n 2 , ... , n l ≤ x d k (n 1 2 + n 2 2 + ⋯ + n l 2) , where n 1 , n 2 , ... , n l ∈ Z + , k ≥ 4 and l ≥ 3 are integers. Previously only the cases of k = 2 , 3 are studied. [ABSTRACT FROM AUTHOR]

In this paper, we apply the circle method to study the average behavior of the triple divisor function over values of quadratic form n 1 2 + n 2 2 + ⋯ + n l 2 with l ⩾ 3. We improve and generalize previous results. [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]

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]

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]

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 a recent paper, J. Lovejoy and the second author conjectured that ranks for four types of unimodal like sequences satisfy certain inequalities. In this paper, we prove these conjectures asymptotically. For this, we use Wright's Circle Method and analyze the asymptotic behavior of certain general partial theta functions. [ABSTRACT FROM AUTHOR]

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

Let a 1 , ⋯ , a 4 be non-zero integers and n any integer. Suppose that a 1 , ⋯ , a 4 and n satisfy some related conditions. In this paper we prove that (i) if a j are not all of the same sign, then the equation a 1 p 1 + a 2 p 2 2 + a 3 p 3 2 + a 4 p 4 2 = n has prime solutions satisfying max ⁡ { p 1 , p 2 2 , p 3 2 , p 4 2 } ≪ | n | + max ⁡ { | a j | } 14 + ε ; (ii) if all a j are positive and n ≫ max ⁡ { | a j | } 15 + ε , then the equation a 1 p 1 + a 2 p 2 2 + a 3 p 3 2 + a 4 p 4 2 = n is soluble in primes p j . [ABSTRACT FROM AUTHOR]

In this paper, we show that all sufficiently large natural numbers satisfying certain local conditions can be written as the sum of k th powers of Piatetski-Shapiro primes, thereby establishing a variant of Waring–Goldbach problem with primes from a sparse sequence. [ABSTRACT FROM AUTHOR]

In this paper, using a variant of the Hardy–Ramanujan Circle Method originally due to E. Wright, we prove asymptotic formulas for spt-crank of ordinary partitions. [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]

In this short paper, it is proved that every sufficiently large even integer is a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 41 powers of 2. Furthermore, there are at least 3.682% odd integers that can be represented as one square of prime, one cube of prime and one fourth power of prime. [ABSTRACT FROM AUTHOR]

Let λ f ( n ) be the Fourier coefficients of primitive holomorphic cusp forms for SL 2 ( Z ) . In this paper, we study the hybrid problem of ternary quadratic forms and Fourier coefficients of primitive holomorphic cusp form. [ABSTRACT FROM AUTHOR]

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012. [ABSTRACT FROM AUTHOR]

In this paper, we show that every pair of large positive even integers can be represented in the form of a pair of Goldbach–Linnik equations, that is, linear equations in two primes and $k$ powers of two. In particular, $k=34$ powers of two suffice, in general, and $k=18$ under the generalised Riemann hypothesis. Our result sharpens the number of powers of two in previous results, which gave $k=62$, in general, and $k=31$ under the generalised Riemann hypothesis. [ABSTRACT FROM PUBLISHER]

In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein–Ramakrishnan's result about the non-existence of the Siegel zeros for G L ( 2 ) L -functions, which allows us to improve preceding estimates. [ABSTRACT FROM AUTHOR]

Let P + (n) denote the largest prime factor of an integer n. In this paper, we study the asymptotics of sums ∑ p ⩽ x P + (p + h) ⩽ y 1 and ∑ n ⩽ x , P + (n) ⩽ y 1 P + (n + h) ⩽ y 2 1 for x ε ⩽ y , y 1 , y 2 ⩽ x. Our results imply that two conjectures concerning the asymptotics for the above two sums, which were considered by Erdős and Pomerance, hold on average. As an application of the first sum above, we can deduce that there are infinitely many primes p ⩽ x such that P + (p + h) > p 1 − ε for all but o ((log ⁡ x) c 0 ) values of shifts h , where c 0 is a small computable constant. As applications of the second sum above, we consider the averaged version of Erdős–Turán's conjecture and the equation a + b = c. In particular, we show that there exists at least a positive density (>17.505%) of triples of integers { a , b , c } such that a + b = c and c ≪ P + (a) P + (c). This upper bound is stronger than c ≪ (∏ p | a b c p) 1 + ε in the abc -conjecture. [ABSTRACT FROM AUTHOR]

Let ( a i , a j ) = 1 , 1 ≤ i < j ≤ s and s = 2 k + 1 , where a 1 , ⋯ , a s , s and k ≥ 4 are nonzero integers. In this paper, we show that if the diagonal diophantine equation a 1 p 1 k + ⋯ + a s p s k = n is satisfying some necessary conditions, then we have the following results: For any ϵ > 0 , we have (i) if a 1 , ⋯ , a s are not all of the same sign, then the above equation has solutions in primes p j satisfying p j ≪ | n | 1 / k + A 3 ⋅ 2 k − 1 + ϵ , (ii) if a 1 , ⋯ , a s are all positive, then the above equation is solvable in prime p j whenever n ≫ A 3 k ⋅ 2 k − 1 + 1 + ϵ . This result is the general case of the Diophantine equations with Small Prime Variables. [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]

It was proved that, for k = 332 , every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In this paper, we sharpen the value of k to 128. [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]

Bringmann, Mahlburg and Rhoades proved asymptotic formulas for all the even moments of the ranks and cranks of partitions with polynomial error terms. In this paper, motivated by their work, we apply the same method and obtain asymptotics for the two rank moments of overpartitions. [ABSTRACT FROM AUTHOR]

Abstract: A recent paper by Hanusa and Nath states many conjectures in the study of self-conjugate core partitions. We prove all but two of these conjectures asymptotically by number-theoretic means. We also obtain exact formulas for the number of self-conjugate t-core partitions for “small” t via explicit computations with modular forms. For instance, self-conjugate 9-core partitions are related to counting points on elliptic curves over with conductor dividing 108, and self-conjugate 6-core partitions are related to the representations of integers congruent to by , a form with finitely many (conjecturally five) exceptional integers in this arithmetic progression, by an ineffective result of Duke–Schulze-Pillot. [Copyright &y& Elsevier]

Abstract: Asymptotic formulas for the positive moments of rank and crank of partitions were obtained by K. Bringmann and K. Mahlburg recently. Motivated by their works, in this paper, we prove asymptotic formulas for the -ranks and their cumulation functions. Asymptotic inequalities between these combinatorial objects are also discovered. In particular, we show that, for fixed integer and sufficiently large , and where (resp. ) denotes the number partitions of with crank (resp. rank) . [Copyright &y& Elsevier]

Abstract: In this short 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 quadratic equations in prime variables and k powers of 2. In particular, we have . We also consider the similar result for cubic equations. [Copyright &y& Elsevier]

The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate. [ABSTRACT FROM AUTHOR]

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k -- 332. [ABSTRACT FROM AUTHOR]

Abstract: Text: For any integer , we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order . The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=H49W8Wyl47o. [Copyright &y& Elsevier]

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established. [ABSTRACT FROM AUTHOR]

In this short paper, we investigate exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we show that all but O(N213/214) integers subject to the necessary local conditions can be represented as the sum of 11 fifth powers of primes. This improves the previous result. [ABSTRACT FROM AUTHOR]

Let p, with or without subscripts, always denote a prime number. In this paper we are able to establish two localized results on a theorem of Prachar which states that almost all positive even integers n can be written as . As a consequence of one result, we prove additionally that each sufficiently large odd integer N can be represented as with for . [ABSTRACT FROM AUTHOR]

Abstract: It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. In this paper, it is proved that the conjecture is true for at least 2.911% of the positive integers satisfying the necessary conditions. This improves the result 1.5% due to Ren (2003) . [Copyright &y& Elsevier]

In this paper we study the distribution of primes in quadratic progressions on average. Our result improves a previous result of Baier and Zhao in some aspects. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2. [Copyright &y& Elsevier]

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54-70, ] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as where U = N with $ \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} $ and K = 2 k, k ⩾ 2. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, with at most O( N) exceptions, all positive integers n ⩽ N can be written as sums of a cube and four cubes of primes. [ABSTRACT FROM AUTHOR]

Let p be prime numbers. In this paper, it is proved that for any integer k≧5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$. This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1-4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$. This method can also be used for some other similar forms. [ABSTRACT FROM AUTHOR]

The Fourier coefficients of classical modular forms of negative weights have been determined for the case for which F(τ) belongs to a subgroup of the full modular group [9]. In this paper, we determine the Fourier coefficients of generalized modular forms of negative weights using the circle method. [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]