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개요

• 모듈라 형식의 푸리에 계수를 estimate 하기 위한 개념
• $a,b\in \mathbb{Z}$와 소수 $p$에 대하여

\[K(a,b;p)=\sum_{1\leq x\leq p-1}e^{2i\pi (ax+b\bar{x})/p},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p\]

• 더 일반적으로 $a,b,m\in \mathbb{Z}$에 대하여

\[K(a,b;m)=\sum_{1\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+b\bar{x})/m}, \quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } m\]

메모

• http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
• Kloosterman, H. D. On the representation of numbers in the form ax² + by² + cz² + dt², Acta Mathematica 49 (1926), pp. 407-464

관련된 항목들

• 타원 모듈라 j-함수 (elliptic modular function, j-invariant)

사전 형태의 자료

• http://en.wikipedia.org/wiki/Kloosterman_sum

관련논문

• Kowalski, E., Ph Michel, and W. Sawin. “Bilinear Forms with Kloosterman Sums and Applications.” arXiv:1511.01636 [math], November 5, 2015. http://arxiv.org/abs/1511.01636.
• Ahlgren, Scott, and Nickolas Andersen. “Kloosterman Sums and Maass Cusp Forms of Half Integral Weight for the Modular Group.” arXiv:1510.05191 [math], October 17, 2015. http://arxiv.org/abs/1510.05191.
• Burkhardt, Paula, Alice Zhuo-Yu Chan, Gabriel Currier, Stephan Ramon Garcia, Florian Luca, and Hong Suh. ‘Visual Properties of Generalized Kloosterman Sums’. arXiv:1505.00018 [math], 30 April 2015. http://arxiv.org/abs/1505.00018.