디리클레 L-함수(Dirichlet L-Function)의 디리클레 급수(Dirichlet Series) 형식은,
는 디리클레 지표
디리클레 L-함수는 다른 L-함수계열처럼 가산(덧셈)과 곱셈의 수학적 상관관계를 직접적으로 보여주는 리만 제타 함수를 근간으로 하는 특수 함수이다.
소수는 리만 제타 함수에서 보여지듯이 가산과 곱셈사이의 연결을 이해하는 중요한 정보이다.
디리클레가 무한히 많은 소수들이 포함되어 있음을 증명하는 디리클레 등차수열 정리에서 디리클레 L-함수(Dirichlet L-Function)를 사용했다.
![{\displaystyle \beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaDoOygdBzjo2ntGNaNe5yjw4ntG5aAo5zjlDoAdEaNw3yqs2aAnE)
![{\displaystyle \beta (s)=L_{-4}(s)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81z2dFoNJDajFCz2dFytGNzNa1ngi0zNKQaAoNajK4yjhBaNe0nDrD)
![{\displaystyle L_{-4}(1)={1 \over 4}\pi }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnDCOzji3zNhCytnEoNBDzgw1zqzFaNw3njlEyta2njo1ajw5ngrA)
![{\displaystyle \zeta (s)=L_{+1}(s)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aje0aAo0zga3otGNoqrAzAw5aNmPngdBajK4yqw4zDe0nqi5zAe5)
![{\displaystyle L_{+1}(1)=\infty }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Po2w5zDm4oNzAzNs2oqo1nDm3zNm4zNKPaDK0nAe3oDzCyjvFo2oQ)
![{\displaystyle \lambda (s)=\sum _{n=0}^{\infty }{{1} \over {(2n+1)^{s}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ntC0zDFByjlFaDJCzjC1yjlDoNaNoNC3atBFz2o3yjJFnjK0nqe3)
리만 제타 함수
따라서,
![{\displaystyle {{\lambda (s)} \over {\zeta (s)}}=(1-2^{-s})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzgwNaqw4ntFCaDdAajs1zNwPzAa0ajBCoDrEzAhEzAa5ngs2oNKN)
![{\displaystyle {{\zeta (s)} \over {\lambda (s)}}={{1} \over {(1-2^{-s})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pnts4nAo4ztiOyqwQoDBAajeNz2zFntBDzthBztiQaNo5ntGQaAzE)
![{\displaystyle {\zeta (s)}={{\lambda (s)} \over {(1-2^{-s})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nDm0oNBCntlEztmPaDzEntaOytdCaNdCngvFnjBEoNrCntFEaqoO)
따라서,
![{\displaystyle L_{+1}(s)=\zeta (s)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzjC2zgw3ntKOa2hBzDoOotBAaDe4zqw1ntC1z2s5zDlCzgo4oNGQ)
![{\displaystyle L_{+1}(s)=\zeta (s)={{\lambda (s)} \over {(1-2^{-s})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoDJDoAi4z2o3atvCzNKNoqhCoAa4zDs0aje5a2wOyqo0otdAztm4)
![{\displaystyle L_{+1}(1)=\infty }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Po2w5zDm4oNzAzNs2oqo1nDm3zNm4zNKPaDK0nAe3oDzCyjvFo2oQ)
모듈러 함수와 푸리에 급수와의 연관성[1]
![{\displaystyle f(\tau )=\sum _{n=0}^{\infty }c(n)e^{2\pi i\tau n}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoqoQaNJFatC3oqzBajzFoNi4agiOyta4zAePngo3oDoPytGOagw1)
![{\displaystyle \quad =c(0)+\sum _{n=1}^{\infty }c(n)e^{2\pi i\tau n}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zqeOnDzCyqhAnqaOoDrEaNFCo2s1aNlCoDmPztmOzNJDatFAoNnC)
![{\displaystyle f\left({{a\tau +b} \over {c\tau +d}}\right)=(c\tau +d)^{k}f(\tau )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BatKQzqaPoNC1age0zArDnqzFzjnDytvDoAe2zgi4ztwQoNmQaNa2)
![{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81otrBnDJFaDC1aqiOztC5ygeQyts2ngvCzqvDzNa5aDoNnja3otvE)
모듈러 군의 감마
멤버이다.
![{\displaystyle L_{-8}(1)={\pi \over 2{\sqrt {2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzqhAaNvDoDs3aAnDnqzFo2s2aNiQo2hEaAnFytK4o2sOntrBatCN)
![{\displaystyle L_{-7}(1)={\pi \over {\sqrt {7}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnAe3atnDotFBnja5yjrAajJEnDeNoNzFnqw0ajKPaAsPagaQa2e3)
![{\displaystyle L_{-3}(1)={1 \over 9}{\pi {\sqrt {3}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zjhFyjzCzDK3nDKQyqw1agvFoNiPatBAyta0yqs4zjFFo2hDaAsQ)
![{\displaystyle L_{+5}(1)={2 \over 5}{\sqrt {5}}\ln \phi }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oNvAajJBzNa2zDw5ajrFnqzEyqrAzjs2ajm5oDGOnqsQnDGPateO)
![{\displaystyle L_{+8}(1)={{\ln(1+{\sqrt {2}})} \over {\sqrt {2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzjC3nqi4ztKNnjG1aga5yjFEyge2njs0nqiPnAi4ntBCoDFAaNa4)
![{\displaystyle L_{+12}(1)={{\ln(2+{\sqrt {3}})} \over {\sqrt {3}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nji4oDw5zDo4atzEztlCzgsPztm5nta2ytCOnte1njFBzAwNzDs2)
![{\displaystyle L_{+13}(1)={2 \over {\sqrt {13}}}\ln \left({{3+{\sqrt {13}}} \over {2}}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PoNs0nqs3oDwNaNe2atGOoDhEaDFBytCQzjFAntJAoNoNoDBBoAi0)
![{\displaystyle }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnDvCo2w2ate3zDaOygo3oNJBoNrEngaNytBEzAsPatiPzgiOoNo4)
![{\displaystyle L_{-4}(2)=K}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aAe3nto4zAnFnghAyjFDotiOz2e4aAiNzDeOzDnAzjlCaNJCa2i3)
![{\displaystyle L_{-3}(2)={1 \over 9}\left(\psi _{1}{\left({1 \over 3}\right)}-\psi _{1}{\left({2 \over 3}\right)}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aAePajnFaNw4aqeQatGOnDs2zgvEatmNagnFaNwPoqw5yjaQoDJE)
![{\displaystyle L_{+1}(2)={1 \over 6}\pi ^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pz2iQngdCyqiNaqvDyjG1aDvDyqe0otw3zji0zNFEnqaQaNJEoqhF)
: 카탈랑 상수,
: 트리감마 함수
- Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.
- ↑ Hecke, E. "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung." Math. Ann. 112, 664-699, 1936