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.
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
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
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
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.
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
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
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
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
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
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
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
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
基金Supported by NSF of China(10671155)Supported by SF of Education Department of Shannxi Province(08JK291)
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant No.11971476)。
文摘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.
基金Supported by National Natural Science Foundation of China
文摘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.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371290 and 11701549)Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2017JM1045)
文摘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.
文摘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.
基金supported partially by Doctoral Research Initiation FundProjectof PanzhihuaUniversity(bkqj2019050).
文摘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.
基金Supported by the Ministry of Science and Technological Development of Serbia(Grant No.174006)
文摘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）.
基金supported by the National Natural Science Foundation of China(Nos.11371195,11471017)the Youth Foundation of Mathematical Tianyuan of China(No.11126302)the Project of Graduate Education Innovation of Jiangsu Province(No.CXZZ12-0381)
文摘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.
基金partially supported by the Templeton Religion Trust under(Grant No.TRT 0159)supported by the Chinese Academy of Sciences and the World Academy of Sciences for CAS-TWAS fellowship
文摘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.
文摘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.
基金Supported by National Natural Science Foundation of China (Grant No. 10771103)
文摘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.
基金supported by the National Natural Science Foundation of China(21427811,21675151)starting support from Ocean University of China。
文摘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.
文摘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.
