In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal ...In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.展开更多
The main purpose of this paper is, using the mean-value theorem of Dirichletl-functions, to study the distribution properties of the hybrid mean value involving certain Hardysums and Ramanujan sum, and give four inter...The main purpose of this paper is, using the mean-value theorem of Dirichletl-functions, to study the distribution properties of the hybrid mean value involving certain Hardysums and Ramanujan sum, and give four interesting identities.展开更多
Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency n...Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, the traditional method for calculating RS values is rather complex since it requires two numbers＇ factorization in two arithmetic functions. For a length-n vector, its Ramanujan-Fourier transform usually involves a series of RS values which will occupy O（n2） memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O（n）. Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O（n2） to O（n In（In（n））） .展开更多
Borwein and Choi conjectured that a polynomial P（x） with coefficients ±1 of degree N - 1 is cyclotomic iffP（x）=±Φp1（±x）ΦP2（±x^p1）…Φpr（±x^p1p2…pr-1）,where N = P1P2 … pτ and the...Borwein and Choi conjectured that a polynomial P（x） with coefficients ±1 of degree N - 1 is cyclotomic iffP（x）=±Φp1（±x）ΦP2（±x^p1）…Φpr（±x^p1p2…pr-1）,where N = P1P2 … pτ and the pi are primes, not necessarily distinct. Here Φ（x） ：= （x^p - 1）/（x - 1） is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the coniecture.展开更多
基金Supported by the National Research and Development Program of China(Grant No.2018YFB1107402)。
文摘In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.
基金This work is supported by the NSF(10271093)the PNSF of P.R.China
文摘The main purpose of this paper is, using the mean-value theorem of Dirichletl-functions, to study the distribution properties of the hybrid mean value involving certain Hardysums and Ramanujan sum, and give four interesting identities.
基金National Natural Science Foundation of China(No.61071070)the Specialized Research Fund for the Doctoral Program of High Education of China(No.20091102120010)
基金Supported by the National Natural Science Foundation of China(No.61071070)
文摘Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, the traditional method for calculating RS values is rather complex since it requires two numbers＇ factorization in two arithmetic functions. For a length-n vector, its Ramanujan-Fourier transform usually involves a series of RS values which will occupy O（n2） memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O（n）. Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O（n2） to O（n In（In（n））） .
基金Research partially supported by Program for New Century Excellent Talents in University Grant # NCET-06-0785by SRF for ROCS, SEM
文摘Borwein and Choi conjectured that a polynomial P（x） with coefficients ±1 of degree N - 1 is cyclotomic iffP（x）=±Φp1（±x）ΦP2（±x^p1）…Φpr（±x^p1p2…pr-1）,where N = P1P2 … pτ and the pi are primes, not necessarily distinct. Here Φ（x） ：= （x^p - 1）/（x - 1） is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the coniecture.
