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

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 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]

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]

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]

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]

J E Littlewood (1885-1977) was a British mathematician well known for his joint work with G H Hardy on Waring's problem and the development of the circle method. In the first quarter of the 20th century, they created a school of analysis considered the best in the world. Littlewood firmly believed that research should be offset by a certain amount of teaching. In this exposition, we will highlight several notable results obtained by Littlewood in the area of number theory. [ABSTRACT FROM AUTHOR]

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.

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 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 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 ( 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]

Let $ N $ be a sufficiently large integer. In this paper, it is proved that, with at most $ O\big(N^{4/9+\varepsilon}\big) $ exceptions, all even positive integers up to $ N $ can be represented in the form $ p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3 $, where $ p_1, p_2, p_3, p_4, p_5, p_6 $ are prime numbers.

Round­robin tournaments are widely used in sports leagues such as football and baseball leagues. The round­robin tournament scheduling problem is one of the most well­known problems in sports scheduling. When creating a schedule for a round­robin tournament, various factors are considered such as travel distance of teams and the number of breaks. Among such factors, the carry­over effect is often considered to evaluate a schedule from the viewpoint of fairness. In this paper, we propose a metaheuristic algorithm for the problem of minimizing the carry-over effect value. For a complete graph whose edges describe all possible matches in a round­robin tournament, we consider a rainbow perfect matching, which we call a base matching, to describe a round of a schedule. By using the circle method, a round­robin tournament schedule can be generated from any base matching. Then we design a metaheuristic algorithm based on iterated local search to search for a “good” base matching, that is, one that generates a round­robin tournament schedule with a low carry­over effect value. We confirmed through computational experiments that our proposed algorithm obtained solutions with carry­over effect values equal to or lower than the best­known values for 15 out of the 17 tested instances with up to 40 teams. In addition, we updated the best­known records for 2 instances.

Consider the maximal operator ... where d is a positive integer, K a Calderón-Zygmund kernel and n ≥ 1. This is a discrete analogue of a real-variable operator studied by Stein and Wainger. The nonlinearity of the phase introduces a variety of new difficulties that are not present in the real-variable setting. We prove ℓ²(ℤn)-bounds for C, answering a question posed by Lillian Pierce. [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]

Abstract: The PresCrIbed suRface Curvature distribution bLade dEsign (CIRCLE) method, proposed by T Korakianitis, is a direct method for designing (or redesigning) high efficiency turbomachinery and fan blades, and isolated airfoils. It was initially introduced to improve the design of high efficiency turbomachinery blades. It is now extended for use with 2D and 3D turbomachinery blades and isolated airfoils. The connection of the profile's leading edge (LE) to the trailing edge (TE), on both sides, is defined using continuity in the surface curvature and its derivatives. Improvements to the aerodynamic performance of the Eppler airfoil are presented in this paper. Improved geometries with continuous curvature have been produced using the CIRCLE method. The performances of Eppler and the redesigned A5 and A6 have been studied using CFD analysis. They are analyzed considering low subsonic flow condition (Reynolds number~105). These are compared to a previous airfoil A4. The redesigned blades’ LE proved favorable as they succeeded in removing pressure ‘spikes’ on the suction side. Comparative analysis with the original results showed significant aerodynamic improvements. Airfoils A4, A5 and A6 have comparatively thicker TE section compared to Eppler. Despite the changes made to the geometry and continuous curvature distribution in A4, A5 and A6 (from Eppler), there appear a separated region on the suction side at about 60% of the chord downstream from the LE. The new blades exhibit higher aerodynamic efficiencies, in terms of overall lift-to-drag ratio up to 40% than the original. The investigations are perfo med at on and off design conditions. [Copyright &y& Elsevier]

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]

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]

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]

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 prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers. [Copyright &y& Elsevier]

In this paper we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions. Using these results, we show that N ‾ (a , c , n) , the number of overpartitions of n with rank congruent to a modulo c , is equidistributed with respect to 0 ≤ a < c , for any c ≥ 2 , as n → ∞ and, in addition, we prove some inequalities between ranks of overpartitions conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018) for n = 6 and n = 10. [ABSTRACT FROM AUTHOR]

In this paper, we obtained that when k = 455, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of unlike powers of primes and k powers of 2.

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

For any integer K≥1 let s(K) be the smallest integer such that in any colouring of the set of squares of the integers in K colours every large enough integer can be written as a sum of no more than s(K) squares, all of the same colour. A problem proposed by Sárközy asks for optimal bounds for s(K) in terms of K. It is known by a result of Hegyvári and Hennecart that s(K)≥Kexp(log2+o(1))logKloglogK. In this article we show that s(K)≤Kexp(3log2+o(1))logKloglogK. This improves on the bound s(K)≪ϵK2+ϵ, which is the best available upper bound for s(K). [ABSTRACT FROM AUTHOR]

Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3/2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).