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On Some Sums Involving Small Arithmetic Functions
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作者 Wen Guang ZHAI 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2024年第10期2497-2518,共22页
Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In thi... Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In this paper,we shall study the mean square of E_(f)(x)and establish some new results of E_(f)(x)for some special functions. 展开更多
关键词 Small arithmetic function exponential sum asymptotic formula
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Operator algebras associated with multiplicative convolutions of arithmetic functions 被引量:2
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作者 Aiju Dong Linzhe Huang Boqing Xue 《Science China Mathematics》 SCIE CSCD 2018年第9期1665-1676,共12页
The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed... The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l^2(N)), the set of all bounded operators on l^2(N).Moreover, the differential operator on l^2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) > 1}. Interesting problems concerning are discussed. 展开更多
关键词 natural numbers arithmetic functions C^*-algebra von Neumann algebra differential operator
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Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain
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作者 Long Chen Zongbing Lin Qianrong Tan 《Algebra Colloquium》 SCIE CSCD 2022年第3期527-540,共14页
Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that ... Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008. 展开更多
关键词 divisor chain integer-valued arithmetic function integer matrix DIVISIBILITY
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Characterization of Arithmetic Functions that Preserve the Sum-of-squares Operation
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作者 Bojan BAI 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2014年第4期689-695,共7页
We characterize all functions f : N → C such that f(m^2 + n^2) = f(m)^2 + f(n)^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (... We characterize all functions f : N → C such that f(m^2 + n^2) = f(m)^2 + f(n)^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (subject to some restrictions on when the choice of the sign is possible) and f(n) =±l/2(again subject to some restrictions on when the choice of the sign is possible). 展开更多
关键词 arithmetic function functional equation sum of squares
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Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions
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作者 Kibrom G.EBREMESKEL Lin Zhe HUANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2019年第8期1300-1310,共11页
In this paper, we consider a multiplicative convolution operator Mf acting on a Hilbert spaces l^2(N,ω;). In particular, we focus on the operators M1 and Mμ, where μ, is the Mobius function. We investigate conditio... In this paper, we consider a multiplicative convolution operator Mf acting on a Hilbert spaces l^2(N,ω;). In particular, we focus on the operators M1 and Mμ, where μ, is the Mobius function. We investigate conditions on the weight ω under which the operators M1 and Mμ are bounded. We show that for a positive and completely multiplicative function f,M1 is bounded on l^2(N, f^2) if and only if ||f||1 <∞, in which case ||M1||2,ω=||f||1, where ωn = f^2(n). Analogously, we show that Mμ is bounded on l^2(N, 1/n^2α) with ||M1||2,ω=ζ(α)/ζ(2α),where ωn= 1 /n^2α,α> 1. As an application, we obtain some results on the spectrum of M1^*M1 and M^*μMμ. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered. 展开更多
关键词 arithmetic functions Mobius function von Neumann algebra
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Dynamics of the Arithmetic Function Ω_k
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作者 石莹 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2008年第4期886-890,共5页
In this paper, we generalize the results of Goldring W. in 2006 and study dynamics of the arithmetic function Ωk.
关键词 DYNAMICS the largest prime factor arithmetic function.
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A New Compound Function and Its Mean Value
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作者 MA Jin-ping GE Jian 《Chinese Quarterly Journal of Mathematics》 CSCD 2010年第2期312-316,共5页
The main purpose of this paper is using the analytic method to study the mean value properties of the arithmetical functions δk((m, n)), δk([m,n]/m),and give several interesting asymptotic formulae for them.
关键词 arithmetic function mean value asymptotic formula
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NOTES ON ERDOS' CONJECTURE
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作者 孔繁超 唐启鹤 《Acta Mathematica Scientia》 SCIE CSCD 2000年第4期533-541,共9页
Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The aut... Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0. 展开更多
关键词 Erdos' conjecture additive arithmetic function sums of independent random variables essential convergence weak convergence
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General divisor functions in arithmetic progressions to large moduli 被引量:1
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作者 WEI Fei XUE BoQing ZHANG YiTang 《Science China Mathematics》 SCIE CSCD 2016年第9期1663-1668,共6页
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
关键词 divisor function arithmetic progression to large moduli mean-value estimate
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Dynamics of a Function Related to the Primes
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作者 Ying SHI Quanhui YANG 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2015年第1期81-90,共10页
Let n = p1p2 ··· pk, where pi(1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p1 + p2)P(p2 + p3) ··· P(pk-1+ pk)P(pk+ p1), where P(n) is the largest ... Let n = p1p2 ··· pk, where pi(1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p1 + p2)P(p2 + p3) ··· P(pk-1+ pk)P(pk+ p1), where P(n) is the largest prime factor of n. Define w0(n) = n and wi(n) = w(wi-1(n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such thatΩs k(n) = Ωs+t k(n) is called the index of periodicity of n. The authors investigate the index of periodicity of n. 展开更多
关键词 DYNAMICS The largest prime factor arithmetic function
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Powers of General Digital Sums
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作者 Feng Juan CHEN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第6期1133-1138,共6页
Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 + a1m0+a2m0m1+…+atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠... Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 + a1m0+a2m0m1+…+atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi(0)=0.write S(n)=i=0∑tfi(ai).In this paper, we give the asymptotic formula for x^-1∑n≤xS(n)^k,where k is a positive integer. 展开更多
关键词 digital sum arithmetic function asymptotic formula
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Engineering DNA logic systems with non-canonical DNA-nanostructures:basic principles,recent developments and bio-applications
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作者 Daoqing Fan Jun Wang +2 位作者 Jiawen Han Erkang Wang Shaojun Dong 《Science China Chemistry》 SCIE EI CSCD 2022年第2期284-297,共14页
Engineering DNA logic systems is considered as one of the most promising strategies for next-generation molecular computers.Owing to the inherent features of DNA,such as low cost,easy synthesis,and controllable hybrid... Engineering DNA logic systems is considered as one of the most promising strategies for next-generation molecular computers.Owing to the inherent features of DNA,such as low cost,easy synthesis,and controllable hybridization,various DNA logic devices with different functions have been developed in the recent decade.Besides,a variety of logic-programmed biological applications are also explored,which initiates a new chapter for DNA logic computing.Although this field has gained rapid developments,a systematical review that could not only elaborate the logic principles of diverse DNA logic devices but also outline recent representative works is urgently needed.In this review,we first elaborate the general classification and logical principle of diverse DNA logic devices,in which the operating strategy of these devices and representative examples are selectively presented.Then,we review state-of-the-art advancements in DNA computing based on different non-canonical DNA-nanostructures during the past decade,in which some classical works are summarized.After that,the innovative applications of DNA computing to logic-controlled bioanalysis,cell imaging,and drug load/delivery are selectively presented.Finally,we analyze current obstacles and suggest appropriate prospects for this area. 展开更多
关键词 DNA logic systems molecular logic non-canonical DNA-nanostructures arithmetic/non-arithmetic functions logical bio-applications
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On the Equation n1n2= n3n4 Restricted to Factor Closed Sets
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作者 San Ying SHI Michel WEBER 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2018年第10期1517-1530,共14页
We study the number of solutions N(B,F) of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^... We study the number of solutions N(B,F) of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^ε1…pk^εk ,εj∈ {0, 1}, 1≤ j ≤ k}, p1,… ,pk being distinct prime numbers. 展开更多
关键词 Diophantine equation arithmetical functions factor closed set
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An Integral Representation for the Weighted Geometric Mean and Its Applications
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作者 Feng QI Xiao Jing ZHANG Wen Hui LI 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2014年第1期61-68,共8页
By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show ... By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality. 展开更多
关键词 Integral representation Cauchy's integral formula arithmetic mean geometric mean weighted arithmetic-geometric mean inequality complete Bernstein function new proof application
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