Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(...Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].展开更多
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)at...Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.展开更多
Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2))...Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.展开更多
Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for conse...Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant sub-sequences,we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences.We show that this ratio is solely dependent on the order of the grid,providing a concise and splendid identity.展开更多
Erdős-Kac定理是数论中的一个经典结果,它描述了在自然数范围内,整数的不同素因子个数的分布渐进服从正态分布。本文主要目的是将Erdős-Kac定理在高斯域中进行推广,令K是高斯域,OK是其整数环。设a∈OK,ω(a)表示其不同的素因子个数,τk...Erdős-Kac定理是数论中的一个经典结果,它描述了在自然数范围内,整数的不同素因子个数的分布渐进服从正态分布。本文主要目的是将Erdős-Kac定理在高斯域中进行推广,令K是高斯域,OK是其整数环。设a∈OK,ω(a)表示其不同的素因子个数,τk(a)是高斯域上k重除数函数。我们用围道积分法,推导出ω(a)的加权均值和m阶中心矩,并由此推导出高斯域上权重为τk(a)的Erdős-Kac定理。这一结果不仅丰富了数论中的分布理论,也为进一步研究高斯域中的数论问题提供了新的工具和方法。The Erdős-Kac theorem is a classical result in number theory, which describes that the distribution of the number of distinct prime factors of integers asymptotically follows a normal distribution. The primary aim of this paper is to extend the Erdős-Kac theorem to Gaussian fields. Let Kbe a Gaussian field and OKbe its ring of integers. Let a∈OK, and ω(a)denote the number of distinct prime factors of a. Let τk(a)be the -fold divisor function on the Gaussian field. Using the method of contour integration, we derive the weighted mean and the k-th central moment ofω(a), and from these, we deduce a weighted form of the Erdős-Kac theorem on Gaussian fields with weight τk(a). This result not only enriches the distribution theory in number theory but also provides new tools and methods for further research on number-theoretical problems in Gaussian fields.展开更多
在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ...在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ=SL(2,Z)上权为偶数k的Hecke特征型,λf n是其在尖点∞处傅里叶展开的第n个标准化傅里叶系数.展开更多
基金Supported by NSFC (Nos.12471009,12301006,12001047,11901566)Beijing Natural Science Foundation (No.1242003)National Training Program of Innovation and Entrepreneurship for Undergraduates(No.202307011)。
文摘Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
文摘Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.
基金supported by the Talent Fund of Beijing Jiaotong University(No.2020RC012)NSFC(No.11871295),supported by NSFC(No.11971476),supported by NSFC(No.12071421)。
文摘Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.
基金Supported by the National Natural Science Foundation of China(Grant No.12471298)the Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.23JSQ031)the Shaanxi Province College Student Innovation and Entrepreneurship Training Program(Grant Nos.S202210699481 and S202310699324X).
文摘Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant sub-sequences,we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences.We show that this ratio is solely dependent on the order of the grid,providing a concise and splendid identity.
文摘Erdős-Kac定理是数论中的一个经典结果,它描述了在自然数范围内,整数的不同素因子个数的分布渐进服从正态分布。本文主要目的是将Erdős-Kac定理在高斯域中进行推广,令K是高斯域,OK是其整数环。设a∈OK,ω(a)表示其不同的素因子个数,τk(a)是高斯域上k重除数函数。我们用围道积分法,推导出ω(a)的加权均值和m阶中心矩,并由此推导出高斯域上权重为τk(a)的Erdős-Kac定理。这一结果不仅丰富了数论中的分布理论,也为进一步研究高斯域中的数论问题提供了新的工具和方法。The Erdős-Kac theorem is a classical result in number theory, which describes that the distribution of the number of distinct prime factors of integers asymptotically follows a normal distribution. The primary aim of this paper is to extend the Erdős-Kac theorem to Gaussian fields. Let Kbe a Gaussian field and OKbe its ring of integers. Let a∈OK, and ω(a)denote the number of distinct prime factors of a. Let τk(a)be the -fold divisor function on the Gaussian field. Using the method of contour integration, we derive the weighted mean and the k-th central moment ofω(a), and from these, we deduce a weighted form of the Erdős-Kac theorem on Gaussian fields with weight τk(a). This result not only enriches the distribution theory in number theory but also provides new tools and methods for further research on number-theoretical problems in Gaussian fields.
文摘在解析数论领域,探讨全纯尖形式的傅里叶系数具有重要的理论价值.通过复变函数的积分技巧,结合自守L-函数的凸界以及积分均值估计方法,研究了不同序列上傅里叶系数的一致均值估计,形式化表达为∑n≤xλf nλf n j,j=2,3,其中f是全模群Γ=SL(2,Z)上权为偶数k的Hecke特征型,λf n是其在尖点∞处傅里叶展开的第n个标准化傅里叶系数.
