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, and due to their completely multiplicative property Are there L-functions other than 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-function of the modular discriminant satisfies the modified relation 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. Ramanujan conjecture Ramanujan conjectured the following: is multiplicative, is not completely multiplicative but for prime and in we have: , and . Ramanujan observed that the quadratic equation of in the denominator of RHS of , would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be and , then which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the , namely for any : In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by . The formulations required to show that it was a consequence were delicate, and not at all obvious. It was the work of Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by . The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the étale cohomology theory were being worked out. Ramanujan–Petersson conjecture for modular forms In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures to the automorphic L-function of the discrete subgroups of . For any modular form one can form the Dirichlet series