The Fourier transform of trigonometric functions on the multiplicative group ${\mathbb Z}^{\times}(m)$
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摘要: 根据乘法群上的傅里叶变换理论框架, 研究了一类三角和, 并揭示了这类三角和与许多数论量 (例如高斯和、 虚二次域类数和伯努利数) 之间的有趣联系.Abstract: Based on the Fourier transform on the multiplicative group
$ {\mathbb Z}^{\times}(m)$ , we study a class of trigonometric sums and reveal interesting connections between these sums and number theoretic quantities, such as Gauss sums, the class number of imaginary quadratic fields, and the Bernoulli number.-
Key words:
- character /
- Fourier transform /
- Gauss sum /
- Kronecker symbol /
- theta function
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[1] BERNDT B C, ZHANG L C. A new class of theta-function identities originating in Ramanujan’s notebook [J]. J Number Theory, 1994, 48: 224-242. [2] LIU Z G. Some Eisenstein series identities related to modular equation of seventh order [J]. Pacific J Math, 2003, 209: 103-130. [3] BOREVICH Z I, SHAFAREVICH I R. Number Theory [M]. New York: Academic Press, 1966. [4] ERDELYI A. Higher Transcendental Functions [M]. New York: McGraw-Hill, 1953. [5] SHEN L C. On the products of three theta functions [J]. Ramanujan J, 1999(3): 343-357.
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