Article
Mathematics
Alessandro Gambini
Summary: The inequality problem with prime variables under certain conditions is studied, and it is proved that the inequality has infinitely many solutions.
Article
Mathematics
Li Zhu
Summary: It is shown that every sufficiently large even integer can be expressed as the sum of prime squares, prime cubes, and powers of 2.
Article
Mathematics, Applied
Yuhui Liu
Summary: This paper proves that for a specific expression R(n), there exists an upper bound O(N4/9+epsilon), indicating that the anticipated asymptotic formula for this expression fails for at most O(N4/9+epsilon) positive integers not exceeding N.
JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Trevor d. Wooley
Summary: This paper investigates a set of non-zero integers and establishes the existence of lines with integral coordinates on a specific affine surface under certain conditions. This conclusion overcomes the traditional convexity barrier and plays an important role in solving the problem.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Libo Li, Zhiwei Hao, Xinru Ding
Summary: In this paper, we introduce the generalized grand Morrey spaces in the framework of probability space setting, and provide the Doob maximal inequalities on these spaces. We also discuss the boundedness of fractional integral operators for regular martingales in this new framework.
JOURNAL OF FUNCTION SPACES
(2022)
Article
Mathematics
Mikhail Dyachenko, Erlan Nursultanov, Sergey Tikhonov, Ferenc Weisz
Summary: This article obtains Fourier inequalities in the weighted L-p spaces for any 1 < p < infinity involving the Hardy-Cesaro and Hardy-Bellman operators. These results are extended to product Hardy spaces for p <= 1. Additionally, the boundedness of the Hardy-Cesaro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. A key tool used is an appropriate version of the Hardy-Littlewood-Paley inequality.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Aleksei Kulikov
Summary: This study presents a contractive Hardy-Littlewood type inequality for functions from H-p(T), with 0 < p <= 2, which is sharp in the first two Taylor coefficients and asymptotically at infinity.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Alan Talmage
Summary: The system of equations has prime solutions for s >= 12, proved using the Hardy-Littlewood circle method and recent results on Vinogradov's mean value theorem. Sufficient conditions for local solvability are provided, determining if the system has solutions modulo each prime.
Article
Mathematics, Applied
Cesar E. Torres Ledesma, Jose Vanterler da C. Sousa
Summary: This paper investigates the Hardy-Littlewood type and the integration by parts results for psi-Riemann-Liouville fractional integrals. It also examines the integration by parts for the psi-Riemann-Liouville and psi-Hilfer fractional derivatives. Additionally, Sobolev-type inequalities involving the psi-Riemann-Liouville and psi-Hilfer fractional derivatives in weighted space are explored.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Xiaosheng Wu, Pengzhen Yang
Summary: According to the Hardy-Littlewood Conjecture, the most likely common differences among primes will increase indefinitely, and every prime can divide all sufficiently large most likely common differences.
COMMUNICATIONS IN MATHEMATICS AND STATISTICS
(2021)
Article
Mathematics, Applied
Helmut Maier, Michael Th Rassias
Summary: This paper proves, on the assumption of the Generalized Riemann Hypothesis, that every sufficiently large odd integer can be represented in a specific form. The proof combines various methods including the infinitude of primes, Piatetski-Shapiro primes, and the Hardy-Littlewood circle method.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics
Feng Liu, Xiao Zhang
Summary: In this paper, the authors establish the bounds for the Hardy-Littlewood maximal operator defined on a finite directed graph over right arrow in the space of bounded p-variation functions. Specifically, they obtain the BVp norms of M-(G) over right arrow for some directed graphs.
Article
Computer Science, Artificial Intelligence
Abd Elhakeem Abd Elnaby, A. H. El-Baz
Summary: This paper introduces a new method using set theory to generate all prime numbers up to a specific natural number, providing an exact formula for determining the number of primes with superior performance compared to traditional sieve methods. The study also includes unified frameworks for well-known sieves and precise closed form expressions for generating primes, showing the effectiveness of the proposed method.
EGYPTIAN INFORMATICS JOURNAL
(2021)
Article
Mathematics, Applied
Sushmita Rawat, Konijeti Sreenadh
Summary: This paper investigates the existence, multiplicity, and regularity of positive weak solutions for the Kirchhoff-Choquard problem. The study shows that each positive weak solution is bounded and satisfies Holder regularity of order s, and the existence of two positive solutions is proved using variational methods and truncation arguments.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Samuel Zbarsky
Summary: In this paper, the centered maximal function associated with centrally symmetric convex bodies is studied, and some conclusions are drawn in specific dimensions. The infimum value problem of having a fixed point is also analyzed under generic shapes.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics
Soumyarup Banerjee
Summary: This paper investigates the sums of four squares of integers with restricted prime factorizations, making progress towards a conjecture by Sun and obtaining generalizations of Gauss and Legendre's results on sums of three squares and Lagrange's four-square theorem.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Felix Baril Boudreau
Summary: This paper presents an algorithm for computing zeta functions of elliptic curves over finite fields and a method for computing the reduction of L-functions of elliptic curves modulo integers. Three main results are obtained, including the extension of a formula for L-functions modulo N to all quadratic twists, a formula for the quotient modulo 2 of L-functions of any two quadratic twists, and the computation of the global root numbers and analytic rank of quadratic twists. The efficiency of different algorithms is compared, and some degree 2 L-functions are computed directly.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Roger C. Baker, Changhao Chen, Igor E. Shparlinski
Summary: We present new estimates for the maximal operator applied to Weyl sums, which are tight in certain cases. We also provide a more detailed analysis of the quadratic case, specifically Gauss sums. In a wide range of parameters, our estimates are optimal and match lower bounds. Our approach is based on combining ideas from Baker (2021) and Chen and Shparlinski (2020).
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Xiumin Ren, Qingqing Zhang, Rui Zhang
Summary: The article introduces a rational quadratic system and provides an upper bound on the number of solutions under specific conditions.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Thomas M. A. Fink
Summary: We introduce and study the recursive divisor function, a recursive analog of the usual divisor function. We give a geometrical interpretation of this function and derive a relation between it and another function. We also explore the problem of ordered factorizations and discover interesting numbers through computation.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Jamie Juul
Summary: We examine the Galois groups of polynomial extensions in a number field and apply the results to study the proportion of primes for which the polynomial has a padic attracting periodic point. The findings are then used in the split case of the Dynamical Mordell-Lang Conjecture.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Pradipto Banerjee, Amit Kundu
Summary: This paper investigates the properties and characteristics of reciprocal polynomials, proves the existence of irreducible reciprocal factors for reciprocal polynomials, and concludes that the irreducible factors of reciprocal polynomials with roots on the unit circle are also reciprocal. This research is significant for the study of reciprocal polynomials.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Solomon Friedberg, David Ginzburg
Summary: This article studies the extension of the classical theta correspondence based on the Weil representation to higher degree metaplectic covers, as well as the existence and periodicity criteria for generic lifts in the resulting theta tower.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Wei Wang
Summary: In this paper, we discuss the algebraic independence of special values of quasi-modular forms and provide a condition for determining when quasi-modular forms are algebraically independent. Moreover, we prove the algebraic independence of values of derivatives of modular forms with algebraic Fourier coefficients based on the theorem proved in the paper.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Tymoteusz Chmiel
Summary: In this paper, the authors numerically compute the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. The entries of this matrix are identified as rational functions in the special values L(f, 1) and L(f, 2) of the corresponding modular form f and one constant. The authors also present related results concerning the rank of the group of period integrals generated by the action of the monodromy group on the conifold period.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Wei Wang, Chuangxun Cheng
Summary: This paper studies the number of distinct prime divisors of the Fourier coefficients of two non-CM newforms with integral coefficients in a probabilistic way. By using the Galois representations attached to the newforms, the effective Chebotarev density theorem, and making assumptions such as the generalized Riemann hypothesis, the authors show that the distribution of the number of distinct primes dividing the Fourier coefficients follows a standard multivariate normal distribution if the newforms are not twists of each other. As a consequence, they prove a result on the multiplicity of modular forms under the generalized Riemann hypothesis.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Jiangwei Xue, Chia -Fu Yu
Summary: This article presents a necessary and sufficient condition for B to be optimally spinor selective for the genus ', generalizing previous criterions for optimal selectivity. It also provides a refinement of the classical trace formula and extends Maclachlan's relative conductor formula for optimal selectivity to all Eichler orders when ' represents a genus of Eichler orders.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
P. G. Walsh
Summary: In this article, the authors further investigated a more general family of curves and discovered a subfamily with surprisingly large average rank, as well as providing a startling example.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Qiyu Yang
Summary: This article investigates large values of the Riemann zeta function within the critical strip and presents a new result that improves upon previous research.
JOURNAL OF NUMBER THEORY
(2024)
Article
Mathematics
Kai (Steve) Fan, Carl Pomerance
Summary: In this article, it is shown that for y <= sqrt(x), Phi(x, y) is strictly less than 0.6x/log(y), except for a few small exceptional cases.
JOURNAL OF NUMBER THEORY
(2024)
