A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers,and f(p)=±1 for each prime p.We prove O-and Ω-results for the summatory function ∑_(n)≤x f...A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers,and f(p)=±1 for each prime p.We prove O-and Ω-results for the summatory function ∑_(n)≤x f(n)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root saving.It is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.展开更多
We consider the mixed arrangement which is composed of the central hyperplane arrangement and a sphere. We discuss the lattice of its intersection set and the relationship between the Mobius function of the mixed arra...We consider the mixed arrangement which is composed of the central hyperplane arrangement and a sphere. We discuss the lattice of its intersection set and the relationship between the Mobius function of the mixed arrangement and the original hyperplane arangement. The Mobius function of the mixed arrangement is equal to the positive or the negative Mobius function of original hyperplane arrangement. Moreover, we give an equality of the chambers and the characteristic polynomial for the mixed arrangement.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
Sarnak’s Disjointness Conjecture states that the Mobius function is disjoint with any zeroentropy flow. This note establishes this conjecture, with a rate, for Furstenberg’s irregular flows on the infinite-dimension...Sarnak’s Disjointness Conjecture states that the Mobius function is disjoint with any zeroentropy flow. This note establishes this conjecture, with a rate, for Furstenberg’s irregular flows on the infinite-dimensional torus.展开更多
For the continuous time Markov chain with transition function P（t） on Z d ＋ , we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P - （t）. If Q, the q-m...For the continuous time Markov chain with transition function P（t） on Z d ＋ , we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P - （t）. If Q, the q-matrix of P（t）, is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegnmnd dual of some Q-function is also given.展开更多
We give an alternative proof of Hua’s theorem that each large N≡5(mod 24)can be written as a sum of five squares of primes.The proof depends on an estimate of exponential sums involving the Mobius function.
基金Supported by(Grant No.12288201)of the National Natural Science Foundation of China。
文摘A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers,and f(p)=±1 for each prime p.We prove O-and Ω-results for the summatory function ∑_(n)≤x f(n)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root saving.It is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.
基金Supported by the National Natural Science Foundation of China(10471020)
文摘We consider the mixed arrangement which is composed of the central hyperplane arrangement and a sphere. We discuss the lattice of its intersection set and the relationship between the Mobius function of the mixed arrangement and the original hyperplane arangement. The Mobius function of the mixed arrangement is equal to the positive or the negative Mobius function of original hyperplane arrangement. Moreover, we give an equality of the chambers and the characteristic polynomial for the mixed arrangement.
基金The research was supported by NSFC(11720101003 and 11801347)key projects of fundamental research in universities of Guangdong Province(2018KZDXM034).
文摘This article traces several prominent trends in the development of Mobius invariant function spaces Q_(K)with emphasis on theoretic aspects.
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Sarnak’s Disjointness Conjecture states that the Mobius function is disjoint with any zeroentropy flow. This note establishes this conjecture, with a rate, for Furstenberg’s irregular flows on the infinite-dimensional torus.
基金Supported by NSFC(Grant Nos.11626245 and 11571043)
文摘For the continuous time Markov chain with transition function P（t） on Z d ＋ , we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P - （t）. If Q, the q-matrix of P（t）, is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegnmnd dual of some Q-function is also given.
文摘We give an alternative proof of Hua’s theorem that each large N≡5(mod 24)can be written as a sum of five squares of primes.The proof depends on an estimate of exponential sums involving the Mobius function.
