带有多个Dirichlet特征和加法特征的Menon-Sury恒等式

陈曼

数学学报 ›› 2019, Vol. 62 ›› Issue (6) : 949-958.

PDF(447 KB)
PDF(447 KB)
数学学报 ›› 2019, Vol. 62 ›› Issue (6) : 949-958. DOI: 10.12386/A2019sxxb0085
论文

带有多个Dirichlet特征和加法特征的Menon-Sury恒等式

    陈曼
作者信息 +

Menon-Sury's Identity with Several Dirichlet Characters and Additive Characters

    Man CHEN
Author information +
文章历史 +

摘要

本文研究了同时带有多个Dirichlet特征和多个加法特征的Menon-Sury恒等式,给出了下列求和的明确表达式
gcd(a1-1,…,as-1,b1,…,brnχ1a1)…χsasλ1b1)…λrbr),
其中n是一个正整数,s,r为非负整数,Zn*是环Zn=Z/nZ的单位群,gcd(,)表示最大公因子,χi(1≤is)是模n的导子为di的Dirichlet特征,λj(1≤jr)是Zn的加法特征.从有限交换群上的Fourier分析的角度看,我们的结果给出了这个算术函数fa1,…,asb1,…,br)=gcd(a1-1,…,as-1,b1,…,brn)在交换群(Zn*s×(Znr上的Fourier展开的系数的明确表达式.

Abstract

This paper studies the Menon-Sury's identity with both Dirichlet characters and additive characters, and we shall give the explicit formula of the following sum
gcd(a1-1,…,as-1,b1,…,br,n)χ1(a1)…χs(as)λ1(b1)…λr(br),
where n is a positive integer, s, r are nonnegative integers, Zn* is the group of units of the ring Zn=Z/nZ, gcd(,) represents the greatest common divisor, χi (1 ≤ is) are Dirichlet characters mod n with conductors di, λj (1 ≤ jr) are additive characters of Zn. From the point of view of Fourier analysis on finite Abelian groups, our result presents the explicit expression of Fourier coefficients of the function f (a1,…,as,b1,…,br)=gcd (a1-1,…,as-1,b1,…,br,n) on the Abelian group (Zn*)s×(Zn)r.

关键词

Menon-Sury恒等式 / Dirichlet特征 / 加法特征 / 中国剩余定理

Key words

Menon-Sury's identity / Dirichlet character / additive character / Chinese remainder theorem

引用本文

导出引用
陈曼. 带有多个Dirichlet特征和加法特征的Menon-Sury恒等式. 数学学报, 2019, 62(6): 949-958 https://doi.org/10.12386/A2019sxxb0085
Man CHEN. Menon-Sury's Identity with Several Dirichlet Characters and Additive Characters. Acta Mathematica Sinica, Chinese Series, 2019, 62(6): 949-958 https://doi.org/10.12386/A2019sxxb0085

参考文献

[1] Apostol T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
[2] Chen M., Hu S., Li Y., On Menon-Sury's identity with several Dirichlet characters, https://arxiv.org/abs/1807.07241.
[3] Haukkanen P., Menon's identity with respect to a generalized divisibility relation, Aequationes Math., 2005, 70(3):240-246.
[4] Haukkanen P., Wang J., High degree analogs of Menon's identity, Indian J. Math., 1997, 39(1):37-42.
[5] Haukkanen P., Wang J., A generalization of Menon's identity with respect to a set of polynomials, Portugal. Math., 1996, 53(3):331-337.
[6] Li Y., Kim D., A Menon-type identity with many tuples of group of units in residually finite Dedekind domains, J. Number Theory, 2017, 175:42-50.
[7] Li Y., Kim D., Menon-type identities derived from actions of subgroups of general linear groups, J. Number Theory, 2017, 179:97-112.
[8] Li Y., Hu X. Y., Kim D., A generalization of Menon's identity with Dirichlet characters, Int. J. Number Theory, 2018, 14:2631-2639.
[9] Li Y., Kim D., Menon-type identities with additive characters, J. Number Theory, 2018, 192:373-385.
[10] Li Y., Hu X. Y., Kim D., A Menon-type identity with multiplicative and additive characters, Taiwanese J. Math., 2019, to appear, https://projecteuclid.org/euclid.twjm/1531382426. In Press.
[11] Menon P. K., On the sum (a-1, n)[(a, n)=1], J. Indian Math. Soc. (N.S.), 1965, 29:155-163.
[12] Miguel C., A Menon-type identity in residually finite Dedekind domains, J. Number Theory., 2016, 164:43-51.
[13] Ramaiah V. S., Arithmetical sums in regular convolutions, J. Reine Angew. Math., 1978, 303/304:265-283.
[14] Sury B., Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo., 2009, 58(2):99-108.
[15] T?rn?uceanu M., A generalization of Menon's identity, J. Number Theory, 2012, 132:2568-2573.
[16] Tóth L., Menon-type identities concerning Dirichlet characters, Int. J. Number Theory, 2018, 14:1047-1054.
[17] Tóth L., Menon's identity and arithmetical sums representing functions of several variables, Rend. Semin. Mat. Univ. Politec. Torino, 2011, 69:97-110.
[18] Tóth L., Short proof and generalization of a Menon-type identity by Li, Hu and Kim, Taiwanse Journal of Mathematics, 2018, to appear, https://projecteuclid.org/euclid.twjm/1537927426. In Press.
[19] Zhao X. P., Cao Z. F., Another generalization of Menon's identity, Int. J. Number Theory, 2017, 13:2373-2379.

PDF(447 KB)

Accesses

Citation

Detail

段落导航
相关文章

/