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In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight : where satisfies : when is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms. == Ramanujan L-function == The Riemann zeta function and the Dirichlet L-function satisfy the Euler product, :\cdots\right)|}} and due to their completely multiplicative property : Are there L-functions except for the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-functions of automorphic forms would satisfy the modified relation :\right)^,|}} where is Ramanujan's tau function. The term : is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ramanujan–Petersson conjecture」の詳細全文を読む スポンサード リンク
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