"Non-holomorphic modular forms"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 6번째 줄: | 6번째 줄: | ||
* modularity<math>G_{2} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2} G_{2}(\tau)-\pi i c(c\tau+d)</math> | * modularity<math>G_{2} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2} G_{2}(\tau)-\pi i c(c\tau+d)</math> | ||
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==E2 as a non-holomorphic modular form== | ==E2 as a non-holomorphic modular form== | ||
| 15번째 줄: | 15번째 줄: | ||
* obtaing modularity losing holomorphicity | * obtaing modularity losing holomorphicity | ||
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==Zagier's function== | ==Zagier's function== | ||
| 32번째 줄: | 32번째 줄: | ||
** [[2518886/attachments/1114356|zagier_hirzebruch.pdf]] | ** [[2518886/attachments/1114356|zagier_hirzebruch.pdf]] | ||
* [http://www.springerlink.com/content/lk767l65118h115h/ Sums involving the values at negative integers of L-functions of quadratic characters] | * [http://www.springerlink.com/content/lk767l65118h115h/ Sums involving the values at negative integers of L-functions of quadratic characters] | ||
| − | ** | + | ** Henri Cohen, 1975 |
2020년 12월 28일 (월) 04:23 기준 최신판
weight 2 Eisenstein series
- \(k=1\)인 경우의 아이젠슈타인급수는 위에서 얻은 푸리에 급수를 이용하여 정의\(G_{2}(\tau) = \zeta(2) \left(1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n} \right)\)
- 원래의 정의와 비슷하게 쓰려면 절대수렴하지 않는 급수 다음과 같이 덧셈의 순서를 따름\(G_{2}(\tau) = \frac{1}{2}\sum_{n\neq 0} \frac{1}{n^2}+\frac{1}{2}\sum_{m\neq0}\sum_{n\in\mathbb{Z}} \frac{1}{(m\tau+n)^{2}}\)
- 정규 아이젠슈타인 급수\(E_{2}(\tau) = 1-24\sum_{n=1}^{\infty} \sigma_{1}(n)q^{n}\)
- modularity\(G_{2} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2} G_{2}(\tau)-\pi i c(c\tau+d)\)
E2 as a non-holomorphic modular form
- \(\tau = x+ iy\), \(y > 0 \)에 대하여 다음과 정의된 함수는 모듈라 성질을 가짐\(G^{*}_{2}(\tau) = G_{2}(\tau)-\frac{\pi}{2y}\)\(E^{*}_{2}(\tau) = E_{2}(\tau)-\frac{3}{\pi y}\)
- obtaing modularity losing holomorphicity
Zagier's function
- Hurwitz class numbers
- Cox_on_Hurwitz_class_number.pdf (Cox's book 319p)
- Zagier's paper
- Zagier-Hirzebruch function
- Intersection numbers of curves on Hibert modular surfaces and modular forms of Nebentypus
- function with coefficients as Hurwitz class numbers
- zagier_hirzebruch.pdf
- Sums involving the values at negative integers of L-functions of quadratic characters
- Henri Cohen, 1975